Skip to main content

Full text of "Mathematical geography"

See other formats


>XX'^\X^<^^\,<V 

m\ ^^^*;\\^ V^^^ ^ ^-^^ 




/) 



J7I 




(i[atmll Ittittwaitg SIthrarg 



ROLLIN ARTHUR HARRIS 

MATHEMATICAL 
LIBRARY 

THE GIFT OF 

EMILY DOTY HARRIS 

1919 



iJL -. i^'^^ 







Cornell University Library 
GA12 .J71 



Mathematical geography, by Willis E. Joh 




olin 



3 1924 032 360 301 




Cornell University 
Library 



The original of tliis book is in 
tlie Cornell University Library. 

There are no known copyright restrictions in 
the United States on the use of the text. 



http://www.archive.org/details/cu31924032360301 



MATHEMATICAL 
GEOGRAPHY 



BY 

WILLIS E. JOHNSON, Ph.B. 

t^ICE PRESIDENT AND PROFESSOR OF GEOGRAPHY AND 

SOCIAL SCIENCES, NORTHERN NORMAL AND 

INDUSTRIAL SCHOOL, ABERDEEN, 

SOUTH DAKOTA 




NEW YORK •.• CINCINNATI •.• CHICAGO 

AMERICAN BOOK COMPANY 

So 



g^r^T*' 



Copyright, 1907. 

BY 

WILLIS E. JOHNSON 

Entered at Stationers' Hall, Londtm 
JOHNSON MATH. GBO< 
E-F 2 



/^,E 



PREFACE 



In the greatly awakened interest in the common-school 
subjects during recent years, geography has received a 
large share. The establishment of chairs of geography in 
some of our greatest universities, the giving of college 
courses in physiography, meteorology, and commerce, 
and the general extension of geography courses in normal 
schools, academies, and high schools, may be cited as 
evidence of this growing appreciation of the importance of 
the subject. 

While physiographic processes and resulting land forms 
occupy a large place in geographical control, the earth in 
its simple mathematical aspects should be better under- 
stood than it generally is, and mathematical geography 
deserves a larger place in the literature of the subject than 
the few pages generally given to it in our physical geog- 
raphies and elementary astronomies. It is generally 
conceded that the mathematical portion of geography 
is the most difficult, the most poorly taught and least 
imderstood, and that students require the most help in 
understanding it. The subject-matter of mathematical 
geography is scattered about in many works, and no one 
book treats the subject with any degree of thoroughness, 
cr even makes a pretense at doing so. It is with the 
view of meeting the need for such a volume that this 
work has been imdertaken. 

Although designed for use in secondary schools and for 
teachers' preparation, much material herein organized 



4 PREFACE 

may be used in the upper grades of the elementary school. 
The subject has not been presented from the point of 
view of a httle child, but an attempt has been made to 
keep its scope within the attainments of a student in a 
normal school, academy, or high school. If a very short 
course in mathematical geography is given, or if students 
are relatively advanced, much of the subject-matter may 
be omitted or given as special reports. 

To the student or teacher who finds some portions too 
difficult, it is suggested that the discussions which seem 
obscure at first reading are often made clear by additional 
explanation given farther on in the book. Usually the 
second study of a topic which seems too difficult should be 
deferred until the entire chapter has been read over care- 
fully. 

The experimental work which is suggested is given for 
the purpose of making the principles studied concrete and 
vivid. The measure of the educational value of a labora- 
tory exercise in a school of secondary grade is not found 
in the academic results obtained, but in the attainment of 
a conception of a process. The student's determination 
of latitude, for example, may not be of much value if its 
worth is estimated in terms of facts obtained, . but the 
forming of the conception of the process is a result of 
inestimable educational value. Much time may be wasted, 
however, if the student is required to rediscover the facts 
and laws of nature which are often so simple that to see 
is to accept and understand. 

Acknowledgments are due to many eminent scholars 
for suggestions, verification of data, and other valuable 
assistance in the preparation of this book. 

To President George W. Nash of the Northern Normal 
and Industrial School, who carefully read the entire manu- 



PEEFACE 5 

script and proof, and to whose thorough training, clear 
insight, and kindly interest the author is under deep 
obligations, especial credit is gratefully accorded. While 
the author has not availed himself of the direct assistance 
of his sometime teacher, Professor Frank E. Mitchell, now 
head of the department of Geography and Geology of the 
State Normal School at Oshkosh, Wisconsin, he wishes 
formally to acknowledge his obhgation to him for an 
abiding interest in the subject. For the critical exami- 
nation of portions of the manuscript bearing upon fields 
in which they are acknowledged authorities, grateful 
acknowledgment is extended to Professor Francis P. 
Leavenworth, head of the department of Astronomy of 
the University of Minnesota; to Lieutenant-Commander 
E. E. Hayden, head of the department of Chronometers 
and Time Service of the United States Naval Observatory, 
Washington; to President F. W. McNair of the Michigan 
College of Mines; to Professor Cleveland Abbe of the 
United States Weather Bureau; to President Robert S. 
Woodward of the Carnegie Institution of Washington; to 
Professor T. C. Chamberlin, head of the department of 
Geology of the University of Chicago; and to Professor 
Charles R. Dryer, head of the department of Geography 
of the State Normal School at Terre Haute, Indiana. For 
any errors or defects in the book, the author alone is 
responsible. 



CONTENTS 



CHAPTER I PAGE 

Introddctokt ................... 9 

CHAPTER II 
The Form op the Earth ... 24 

CHAPTER III 
The Rotation or the Earth 45 

CHAPTER IV 
Longitude and Time , . ... 62 

CHAPTER V 
Circumnavigation and Time ... 92 

CHAPTER VI 
The Earth's Revolution ,,,,,,. 104 

CHAPTER VII 
Time and the Calendar 132 

CHAPTER Vni 
Seasons . , , , , . . 146 

CHAPTER IX 
Tides ,176 

CHAPTER X 
Map Projections . . . . 190 

7 



8 CONTENTS 

f 

CHAPTER XI PAGE 

The United States Government Land Survey . . . . . . 226 

CHAPTER XII 

TRIANOni/ATION IN MEASUREMENT AND ScRVBT . 237 

CHAPTER XIII 
The Earth in Space 246 

CHAPTER XIV 
Historical Sketch 268 



Appendix o>.. • 279 

Glossary . 314 

Index .,,,.,.,......c.. .... 323 



MATHEMATICAL GEOGRAPHY 



CHAPTER I 
introductory 

Observations and Experiments 

Observations of the Stars. On the first clear evening, 
observe the " Big Dipper " * and the polestar. In Septem- 
ber and in December, early in the evening, they will be 
nearly in the positions represented in Figure 1. Where 
is the Big Dipper 
later in the evening? 
Find out by observa- 
tions. 

Learn readily to 
pick out Cassiopeia's 
Chair and the Little 
Dipper. Observe 
their apparent mo- 
tions also. Notice 
the positions of stars 
in different portions of the sky and observe where they are 
later in the evening. Do the stars around the polestar 
remain in the same position in relation to each other, — 
the Big Dipper always like a dipper, Cassiopeia's Chair 

* In Ursa Major, commonly called the " Plow," " The Great 
Wagon," or " Charles's Wagon " in England, Norway, Germany, and 
other countries. 



CD 







^ '"■• 


•♦ 


• • 'i< 





■i ■' 


o 

1 


<•"■■■-•■•... / ..•■ 

Jttle D'PPe,' • North Star. *' 


(b 


ij> 


*... 




September 





Fig. I 



10 



INTRODUCTOBY 



always like a chair, and both always on opposite sides of 
the polestar? In what sense may they be called " fixed " 
stars (see pp. 108, 265)? 

Make a sketch of the Big Dipper and the polestar, 
recording the date and time of observation. Preserve 
your sketch for future reference, marking it Exhibit 1. 
A month or so later, sketch again at the same time of 
night, using the same sheet of paper with a common 
polestar for both sketches. In making your sketches 
be careful to get the angle formed by a line through 
the " pointers " and the polestar with a perpendicular to 
the horizon. This angle can be formed by observing the 
side of a building and the pointer line. It can be 
measured more accurately in the fall months with a pair 
of dividers having straight edges, by placing 
one outer edge next to the perpendicular 
side of a north window and opening the 
dividers until the other outside edge is 
parallel to the pointer line (see Fig. 2). 
Now lay the dividers on a sheet of paper 
and mark the angle thus formed, repre- 
senting the positions of stars with asterisks. 
Two permy rulers pinned through the ends 
will serve for a pair of dividers. 
Phases of the Moon. Note the position 
of the moon in the sky on successive nights at the same 
hour. Where does the moon rise? Does it rise at the 
same time from day to day? When the full moon may 
be observed at sunset, where is it? At smirise? When 
there is a full moon at midnight, where is it? Assume 
it is sunset and the moon is high in the sky, how much of 
the hghted part can be seen? 
Answers to the foregoing questions should be based upon 




Fig. a 



THE NOON SHADOW 11 

first-hand observations. If the questions cannot easily 
be answered, begin observations at the first opportunity. 
Perhaps the best time to begin is when both sun and moon 
may be seen above the horizon. At each observation 
notice the position of the sun and of the moon, the portion 
of the Hghted part which is turned toward the earth, and 
bear in mind the simple fact that the moon always shows a 
lighted half to the sun. If the moon is rising when the sun 
is setting, or the sun is rising when the moon is setting, 
the observer must be standing directly between them, or 
approximately so. With the sim at your back in the east 
and facing the moon in the west, you see the moon as 
though you were at the sim. How much of the hghted 
part of the moon is then seen? By far the best apparatus 
for illustrating the phases of the moon is the sun and 
moon themselves, especially when both are observed above 
the horizon. 

The Noon Shadow. Some time early in the term from 
a convenient south window, measure upon the floor the 
length of the shadow when it is shortest during the 
day. Record the measurement and the date and time of 
day. Repeat the measurement each week. Mark this 
Exhibit 2. 

On a south-facing window sill, strike a north-south line 
(methods for doing this are discussed on pp. 61, 130). 
Erect at the south end of this hne a perpendicular board, 
say six inches wide and two feet long, with the edge next 
the north-south line. True it with a- plumb Une; one 
made with a bullet and a thread will do. This should 
be so placed that the shadow from the edge of the board 
may be recorded on the window sill from 11 o'clock, a.m., 
until 1 o'clock, p.m. (see Fig. 3). 

Carefully cut from cardboard a semicircle and mark the 



12 



INTRODUCTORY 




Noon Shadow 

Fig. 3 



degrees, beginning with the 
middle radius as zero. Fasten 
this upon the window sill 
with the zero meridian coin- 
ciding with the north-south 
line. Note accurately the 
clock time when the shadow 
from the perpendicular board 
crosses the line, also where 
the shadow is at twelve 
o'clock. Record these facts 
with the date and preserve as 
Exhibit 3. Continue the ob- 
servations every few days. 
The Sun's Meridian Altitude. When the shadow 

is due north, carefully 

measure the angle formed 

by the shadow and a level 

line. The simplest way is 

to draw the window shade 

down to the top of a sheet 

of cardboard placed very 

nearly north and south with 

the bottom level and then 

draw the shadow line, the 

lower acute angle being the 

one sought (see Fig. 4). 

Another way is to drive a 

pin in the side of the window 

casing, or in the edge of the 

vertical board (Fig. 3) ; fasten a thread to it and connect 

the other end of the thread to a point on the sill where the 

shadow falls. A still better method is shown on p. 172. 




Altitude of Sun at noon" 
Fig. 4 



CENTRIFUGAL FORCE 



13 



Since the shadow is north, the sun is as high in the sky- 
as it will get during the day, and the angle thus measured 
gives the highest altitude of the sun for the day. Record 
the measurement of the angle with the date as Exhibit 4. 
Continue these records from week to week, especially 
noting the angle on one of the following dates: March 21, 
June 22, September 23, December 22. This angle on 
March 21 or September 23, if subtracted from 90° will 
equal the latitude * of the observer. 



A Few Teems Explained 

Centrifugal Force. The hteral meaning of the word 
suggests its current meaning. It comes from the Latin 
centrum, center; and 
jugere, to flee. A cen- 
trifugal force is one 
directed away from a 
center. When a stone 
is whirled at the end of 
a string, the pull which 
the stone gives the 
string is called centri- 
fugal force. Because 
of the inertia of the 
stone, the whirhng 
motion given to it 
by the arm tends to 




Tends to fly off 

Fig. 5 



make it fly off in a straight line (Fig. 5), —and this 
it will do if the string breaks. The measure of the 
centrifugal force is the tension on the string. If the 
string be fastened at the end of a spring scale and the 

* This is explained oa pp. 170, 171. 



14 INTRODUCTORY 

stone whirled, the scale will show the amount of the centri- 
fugal force which is given the stone by the arm that 
whirls it. The amount of this force * (C) varies with the 
mass of the body (m), its velocity (v), and the radius 
of the circle (r) in which it moves, in the following ratio : 

r : 

The instant that the speed becomes such that the avail- 

able strength of the string is less than the value of ' 

r 

however slightly, the stone will cease to follow the curve 
and will immediately take a motion at a uniform speed 
in the straight line with which its motion happened to 
coincide at that instant (a tangent to the circle at the point 
reached at that moment). 

Centrifugal Force on the Surface of the Earth. The 
rotating earth imparts to every portion of it, save along 
the axis, a centrifugal force which varies according to the 
foregoing formula, r being the distance to the axis, or the 
radius of the parallel. It is obvious that on the surface 
of the earth the centrifugal force due to its rotation is 
greatest at the equator and zero at the poles. 

At the equator centrifugal force (C) amounts to about 
2 59 that of the earth's attraction {g), and thus an. object 
there which weighs 288 pounds is lightened just one pound 
by centrifugal force, that is, it would weigh 289 pounds 

were the earth at rest. At latitude 30°, C = —^ (that is, 

* "On the use of symbols, such as C for centrifugal'force, ^ for latitude, 
etc., see Appendix, p. 307. 



CENTRIPETAL FORCE 15 

centrifugal force is '^|^ the force of the earth's attraction); 
at45°,C=^;at60^C = ^^. ' 

For any latitude the " lightening effect " centrifugal force 
due to the earth's rotation equals -|^ times the square of the 

cosine of the latitude (C = -7^ X cos^ <t>). By referring 

289 

to the table of cosines in the Appendix, the student can 

easily calculate the " lightening " influence of centrifugal 

force at his own latitude. For example, say the latitude 

of the observer is 40°. 

Cosme 40° = .7660. -^ X .7660 ' = 7:^ • 
289 492 



Thus the earth's attraction for an object on its surface 
at latitude 40° is 492 times as great as centrifugal force 
there, and an object weighing 491 pounds at that latitude 
would weigh one poimd more were the earth at rest.* 

Centripetal Force. A centripetal {centrum, center; peter e 
to seek) force is one directed toward a center, that is, al 
right angles to the direction of motion of a body. To 
distinguish between centrifugal force and centripetal 
force, the student should realize that forces never occur 
singly but only in pairs and that in any force action there 
are always two bodies concerned. Name them A and B. 
Suppose A pushes or pulls B with a certain strength. 
This cannot occur except B pushes or pulls A by the same 
amount and in the opposite direction. This is only a 
simple way of stating Newton's third law that to every 

* These calculations are based upon a spherical earth and make no 
allowances for the oblateness. 



16 INTRODUCTORY 

action (A on B) there corresponds an equal and opposite 
reaction (B on A). 

Centrifugal force is the reaction of the body against the 
centripetal force which holds it in a curved path, and it 
must always exactly equal the centripetal force. In the 
case of a stone whirled at the end of a string, the necessary 
force which the string exerts on the stone to keep it in a 
curved path is centripetal force, and the reaction of the 
stone upon the string is centrifugal force. 

The formulas for centripetal force are exactly the same 
as those for centrifugal force. Owing to the rotation of 
the earth, a body at the equator describes a circle with 
uniform speed. The attraction of the earth supplies the 
centripetal force required to hold it in the circle. The 
earth's attraction is greatly in excess of that which is 
required, being, in fact, 289 times the amoimt needed. 
The centripetal force in this case is that portion of the attrac- 
tion which is used to hold the object in the circular course. 
The excess is what we call the weight of the body or the 
force of gravity. 

If, therefore, a spring balance suspending a body at the 
equator shows 288 pounds, we infer that the earth really 
pulls it with a force of 289 pounds, but one pound of this 
pull is expended in changing the direction of the motion 
of the body, that is, the value of centripetal force is one 
pound. The body pulls the earth to the same extent, 
that is, the centrifugal force is also one pound. At the 
poles neither centripetal nor centrifugal force is exerted 
upon bodies and hence the weight of a body there is the 
full measure of the attraction of the earth. 

Gravitation. Gravitation is the all-pervasive force by 
virtue of which every particle of matter in the universe 
is constantly drawing toward itself every other particle 



GRAVITATION 



17 



of matter, however distant. The amount of this attrac- 
tive force existing between two bodies depends upon 
(1) the amoimt of matter in them, and (2) the distance 
they are apart. 

There are thus two laws of gravitation. The first law, 
the greater the mass, or amount of matter, the greater the 
attraction, is due to the fact that each particle of matter 
has its own independent attractive force, and the more 
there are of the par- 
ticles, the greater 
is the combined 
attraction. 

The Second Law 
Explained. In gen- 
eral terms the law 
is that the nearer 
an object is, the 
greater is its air 
tractive force. Just 
as the heat and 
light of a flame are 
greater the nearer one gets to it (Fig. 6), because more rays 
are intercepted, so the nearer an object is, the greater is its at- 

H 




More lays interceptbd when near theflame- 

Fig. 6 



F^^a 



B 



Fig. 7 



D 



traction. The ratio of the increase of the power of gravita- 
tion as distance decreases, may be seen from Figures 7 and 8. 



JO. MATH. OBO.— 2 



18 



INTRODUCTORY 



Two lines, AD and AH (Fig. 7), are twice as far apart 
at C as at 5 because twice as far away; three times as far 
apart at Z) as at S because three times as far away, etc. 
Now light radiates out in every direction, so that light 
coming from point A' (Fig. 8), when it reaches B' will be 



H' 





^^ 


'"■^ 


I"-v^_-i-^ 




^"^ 


^ '""-ll 




^ 


0^ 




^^.^^^^^^a^^^w^" 


' '^ 




i. ! 'fes 



b' 



c 



D' 



Fig. a 



spread over the square of B'F'; at C", on the square C'G'; 
at D' on the square B'H', etc. C" being twice as far 
away from A' as B' , the side C'G' is twice that of B'F', 
as we observed in Fig. 7, and its square is four times as 
great. Line D' H' is three times as far away, is three 
times as long, and its square is nine times as great. The 
Ught being spread over more space in the more distant 
objects, it will light up a given area less. The square 
at B' receives all the light within the four radii, the 
same square at C receives one fourth of it, at D' one 
ninth, etc. The amount of light decreases as the square of 
the distance increases. The force of gravitation is exerted 
in every direction and varies in exactly the same way. 
Thus the second law of gravitation is that the force varies 
inversely as the square of the distance. 

Gravity. The earth's attractive influence is called 
gravity. The weight of an object is simply the measure of 



GRAVITY 19 

the force of gravity. An object on or above the surface 
of the earth weighs less as it is moved away from the 
center of gravity.* It is difficult to reahze that what we 
call the weight of an object is simply the excess of attrac- 
tion which the earth possesses for it as compared with 
other forces acting upon it, and that it is a purely relative 
affair, the same object having a different weight in different 
places in the solar system. Thus the same poimd-weight 
taken from the earth to the sun's surface would weigh 27 
pounds there, only one sixth of a pound at the surface of 
the moon, over 2J pounds on Jupiter, etc. If the earth 
were more- dense, objects would weigh more on the surface. 
Thus if the earth retained its present size but contained as 
much matter as the sun has, the strongest man in the world 
could not Uft a silver half dollar, for it would then weigh 
over five tons. A pendulum clock would then tick 575 
times as fast. On the other hand, if the earth were no 
denser than the sun, a half dollar would weigh only a 
trifle more than a dime now weighs, and a pendulum clock 
would tick only half as fast. 

From the table on p. 266 giving the masses and 
distances of the sun, moon, and principal planets, many 
interesting problems involving the laws of gravitation 
may be suggested. To illustrate, let us take the problem 
" What would you weigh if you were on the moon? " 

Weight on the Moon. The mass of the moon, that is, 
the amount of matter in it, is ir that of the earth. 
Were it the same size as the earth and had this mass, one 
pound on the earth would weigh a Irttle less than one 
eightieth of a pound there. According to the first law of 
gravitation we have this proportion: 

1. Earth's attraction : Moon's attraction : : l:iV- 

* For a more accurate and detailed discussion of gravity, see p. 279. 



20 INTBODUCTORT 

But the radius of the moon is 1081 miles, only a little 
more than one fourth that of the earth. Since a person 
on the moon would be so much nearer the center of gravity 
than he is on the earth, he would weigh much more there 
than here if the moon had the same mass as the earth. 
According to the second law of gravitation we have this 
proportion : 

2. Earth's attraction : Moon's attraction : : ^ : _„,2 • 
We have then the two proportions: 

1. Att. Earth : Att. Moon : : 1 : ^. 

1 1 



2. Att. Earth : Att. Moon 



4000' ■ 1081=* 
Combining these by multiplying, we get 

Att. Earth : Att. Moon : : 6 : 1. 

Therefore six pounds on the earth would weigh only 
one pound on the moon. Your weight, then, divided by 
six, represents what it would be on the moon. There 
you could jump six times as high — if you could hve to 
jump at all on that cold and almost airless satellite (see 
pp. 236, 264). 

The Sphere, Circle, and Ellipse. A sphere is a solid 
boimded by a curved surface all points of which are equally 
distant from a point within called the center. 

A circle is a plane figure bounded by a curved line all 
points of which are equally distant from a point -ndthin 
called the center. In geography what we commonly call 
circles such as the equator, parallels, and meridians, are 
really only the circumferences of circles. Wherever used 



THE SPHERE, CIRCLE, AND ELLIPSE 21 

in this book, unless otherwise stated, the term circle 
refers to the circumference. 

Every circle is conceived to be divided into 360 equal 
parts called degrees. The greater the size of the circle, 
the greater is the length of each degree. A radius of a 
circle or of a sphere is a straight line from the boundary 
to the center. Two radii forming a straight line con- 
stitute a diameter. 

Circles on a sphere dividing it into two hemispheres are 
called great circles. Circles on a sphere dividing it into 
unequal parts are called small circles. 

All great circles on the same sphere bisect each other, 
regardless of the angle at which they cross one another. 
Thaj; this may be clearly seen, with a globe before you test 
these two propositions: 

a. A point 180° in any direction from one point in a 
great circle must lie in the same circle. 

b. Two great circles on the same sphere must cross 
somewhere, and the point 180° from the one where they 
cross, hes in both of the circles, thus each great circle 
divides the other into two equal parts. 

An angle is the difference in direction of two* lines which, 
if extended, would meet. Angles are measured by using 
the meeting point as the center of a circle and finding the 
fraction of the circle, or number of degrees of the circle, 
included between the lines. It is well to practice esti- 
mating different angles and then to test the accuracy of 
the estimates by reference to a graduated quadrant or 
circle having the degrees marked. 

An ellipse is a closed plane curve such that the sum of 
the distances from one point in it to two fixed points within, 
called foci, is equal to the sum of the distances from any 
other point in it to the foci. The eUipse is a conic section 



22 



INTRODUCTORY 



formed by cutting a right cone by a plane passing obliquely 
through its opposite sides (see Ellipse in Glossary). 

To construct an ellipse, 
drive two pins at points 
for foci, say three inches 
apart. With a loop of 
non-elastic cord, say ten 
inches long, mark the 
boundary Une as repre- 
sented in Figure 10. 

Orbit of the Earth. 
The orbit of the earth 
is an elUpse. To lay off 
an eUipse which shall 
quite correctly represent 
the shape of the earth's 
orbit, place pins one 




Ellipse. 
A&A, Foci. C D, Minor Axis 
XV. Major Axis. A toA'.Focal 
Distance. AM+AM'AN*AN 

Fig. 9 



tenth of an inch apart and make a loop of string 12.2 inches 

long. This loop 

can easily be made 

by driving two pins 

6.1 inches apart 

and tying a string 

looped around 

them. 

Shape of the 
Earth. The earth 
is a spheroid, or a 
soUd approaching 
a sphere (see Sphe- 
roid in Glossary). 
The diameter upon which it rotates is called the axis. 
The ends of the axis are its poles. Imaginary hues on the 




To Construct 
An El lip 



Fig. 10 



SHAPE OF THE EARTH 23 

surface of the earth extending from pole to pole are called 
meridians.* While any number of meridians may be 
conceived of, we usually think of them as one degree apart. 
We say, for example, the ninetieth meridian, meaning the 
meridian ninety degrees from the prime or initial meridian. 
What kind of a circle is a meridian circle? Is it a true 
circle? Why? 
The equator is a great circle midway between the poles. > 
Parallels are small circles parallel to the equator. 
It is well for the student to bear in mind the fact that 
it is the earth's rotation on its axis that determines most of 
the foregoing facts. A sphere at rest would not have 
equator, meridians, etc. 

* The term meridian is frequently used to designate a great circle 
passing through the poles. In this book such a circle is designated a 
meridian circle, since each meridian is numbered regardless of its oppo- 
site meridian. 



CHAPTER II 

the form of the earth 

The Earth a Sphere 

Circumnavigation. The statements commonly ^ven as 
proofs of the spherical form of the earth would often apply 
as well to a cylinder or an egg-shaped or a disk-shaped 
body. " People have sailed around it," " The shadow of 
the earth as seen in the eclipse of the moon is always cir- 
cular," etc., do not in themselves prove that the earth is a 
sphere. They might be true if the earth were a cylinder 
or had the shape of an egg. " But men have sailed around 
it in different directions." So might they a lemon-shaped 
body. To make a complete proof, we must show that men 
have sailed around it in practically every direction and 
have found no appreciable difference in the distances in 
the different directions. 

Earth's Shadow always Circular. The shadow of the 
earth as seen in the lunar eclipse is always circular. But 
a dollar, a lemon, an egg, or a cylinder may be so placed 
as always to cast a circular shadow. When in addition 
to this statement it is shown that the earth presents many 
different sides toward the sun during different eclipses of 
the moon and the shadow is always circular, we have a 
proof positive, for nothing but a sphere casts a circular 
shadow when in many different positions. The fact that 
eclipses of the moon are seen in different seasons and at 
different times of day is abundant proof that practically 



TELESCOPIC OBSERVATIONS 



25 




F^!. II. Ship's rigging distinct. 
Water hazy. 



all sides of the earth are turned toward the sun during 

different eclipses. 

Almost Uniform Surface Gravity. An object has almost 

exactly the same weight in 

different parts of the earth 

(that is, on the surface), 

showing a practically common 

distance from different points 

on the earth's surface to the 

center of gravity. This is 

ascertained, not by carrying 

an object all over the earth 

and weighing it with a pair 

of spring scales (why not 

balances?), but by noting the 

time of the swing of the 

pendulum, for the rate of its swing varies according 

to the force of gravity. 

Telescopic Observations. If 
we look through a telescope 
at a distant object over a 
level surface, such as a body 
of water, the lower part is 
hidden by the intervening 
curved surface. This has 
been observed in many differ- 
ent places, and the rate of 
curvature seems imiform 
everywhere and in every 
direction. Persons ascending 
in balloons or living on high 

elevations note the appreciably earlier time of sunrise 

or later time of sunset at the higher elevation. 




Fig. n. Water distinct. Rigging 
ill-defined. 



26 THE FORM OF THE EARTH 

Shifting of Stars and Difference in Time. The proof 
which &st demonstrated the curvature of the earth, and 
one which the student should clearly understand, is the 
disappearance of stars from the southern horizon and the 
rising higher of stars from the northern horizon to persons 
traveling north, and the sinking of northern stars and the 
rising of southern stars to south-bound travelers. After 
people had traveled far enough north and south to make 
an appreciable difference in the position of stars, they 
observed this apparent rising and sinking of the sky. Now, 
two travelers, one going north and the other going south, 
wiU see the sky apparently elevated and depressed at the 
same time; that is, the portion of the sky that is rising for 
one will be sinking for the other. Since it is impossible 
that the stars be both rising and sinking at the same 
time, only one conclusion can follow, — the movement of 
the stars is apparent, and the path traveled north and 
south must be curved. 

Owing to the rotation of the earth one sees the same 
stars in different positions in the sky east and west, so the 
proof just given simply shows that the earth is curved in 
a north and south direction. Only when timepieces were 
invented which could carry the time of one place to differ- 
ent portions of the earth could the apparent positions 
of the stars prove the curvature of the earth east and 
west. By means of the telegraph and telephone we 
have most excellent proof that the earth is curved east 
and west. 

If the earth were flat, when it is sunrise at Philadelphia 
it would be sunrise also at St. Louis and Denver. Sun 
rays extending to these places which are so near together 
as compared with the tremendous distance of the sun, over 
ninety miUions of miles away, would be almost parallel 



ACTUAL MEASUREMENT 27 

on the earth and would strike these points at about the 
same angle". But we know from the many daily messages 
between these cities that sun time in Philadelphia is an 
hour later than it is in St. Louis and two hours later than 
in Denver. 

When we know that the curvature of the earth north 
and south as observed by the general and practically 
uniform rising and sinking of the stars to north-bound and 
south-bound travelers is the same as the curvature east 
and west as shown by the difference in time of places 
east and west, we have an excellent proof that the earth is 
a sphere. 

Actual Measurement. Actual measurement in many 
different places and in nearly every direction shows a prac- 
tically uniform ciu-vature in the different directions. In 
digging canals and laying watermains, an allowance must 
always be made for the curvature of the earth; also in 
surveying, as we shaU notice more explicitly farther on. 

A simple rule for finding the amount of curvature for 
any given distance is the following: 

Square the number of miles representing the distance, and 
two thirds of this number represents in feet the departure 
from a straight line. 

Suppose the distance is 1 mile. That number squared 
is 1, and two thirds of that number of feet is 8 inches. 
Thus an allowance of 8 inches must be made for 1 mile. 
If the distance is 2 miles, that number squared is 4, and 
two thirds of 4 feet is 2 feet, 8 inches. An object, then, 
1 mile away sinks 8 inches below the level line, and at 2 
miles it is below 2 feet, 8 inches. 

To find the distance, the height from a level line being 
given, we have the converse of the foregoing rule : 

Midtiply the number representing the height in feet by li, 



28 



THE FORM OF THE EARTH 



and the square root of this product represents the number of 
miles distant the object is situated. 

The following table is based upon the more accurate 
formula : 





Distance 


(miles) 


= 1.317Vheight (feet). 


m. ft. 


Dist. miles 


Ht. ft. 


Dist. miles 


Ht. ft. 


Dist. miles 


1 


1.32 


50 


9.31 


170 


17.17 


2 


1.86 


55 


9.77 


180 


17.67 


3 


2.28 


60 


10.20 


190 


18.15 


4 


2.63 


65 


10.62 


200 


18.63 


5 


2.94 


70 


11.02 


300 


22.81 


6 


3.23 


75 


11.40 


400 


26.34 


7 


3.48 


80 


11.78 


500 


29.45 


8 


3.73 


85 


12.14 


600 


32.26 


9 


3.95 


90 


12.49 


700 


34.84 


10 


4.16 


95 


12.84 


800 


37.25 


15 


5.10 


100 


13.17 


900 


39.51 


20 


5.89 


110 


13.81 


1000 


41.65 


25 


6.59 


120 


14.43 


2000 


58.90 


30 


7.21 


130 


15.02 


3000 


72.13 


35 


7.79 


140 


15.58 


4000 


83.30 


40 


8.33 


150 


16.13 


5000 


93.10 


45 


8.83 


160 


16.66 


Mile 


95.70 



The Earth an Oblate Spheroid 

Richer's Discovery. In the year 1672 John Richer, the 
astronomer to the Royal Academy of Sciences of Paris, 
was sent by Louis XIV to the island of Cayenne to make 
certain astronomical observations. His Parisian clock had 
its pendulum, slightly over 39 inches long, regulated to beat 
seconds. Shortly after his arrival at Cayenne, he noticed 
that the clock was losing time, about two and a half min- 
utes a day. Gravity, evidently, did not act with so much 
force near the equator as it did at Paris. The astronomer 
found it necessary to shorten the pendulum nearly a 
quarter of an inch to get it to swing fast enough. 



AMOUNT OF OBLATENESS 29 

Richer reported these interesting facts to his colleagues 
at Paris, and it aroused much discussion. At first it was 
thought that greater centrifugal force at the equator, 
counteracting the earth's attraction more there than else- 
where, was the explanation. The difference in the force 
of graAdty, however, was soon discovered to be too great 
to be thus accounted for. The only other conclusion was 
that Cayenne must be farther from the center of gravity 
than Paris (see the discussion of Gravity, Appendix, 
p. 279; also Historical Sketch, pp. 273-275). 

Repeated experiments show it to be a general fact that 
pendulums swing faster on the surface of the earth as one 
approaches the poles. Careful measurements of arcs of 
meridians prove beyond question that the earth is flattened 
toward the poles, somewhat Uke an oblate spheroid. 
Further evidence is found in the fact that the sun and 
planets, so far as ascertained, show this same flattening. 

Cause of Oblateness. The cause of the oblateness is 
the rotation of the body, its flattening effects being more 
marked in earlier plastic stages, as the earth and other 
planets are generally believed to have been at one time. 
The reason why rotation causes an equatorial bulging 
is not difficult to imderstand. Centrifugal force increases 
away from the poles toward the equator and gives a lifting 
or lightening influence to portions on the surface. If the 
earth were a sphere, an object weighing 289 pounds at 
the poles would be lightened just one pound if carried to 
the swiftly rotating equator (see p. 280). The form 
given the earth by its rotation is called an oblate spheroid 
or an ellipsoid of rotation. 

Amount of Oblateness. To represent a meridian circle 
accurately, we should represent the polar diameter about 
j-^^ part shorter than the equatorial diameter. That this 



30 THE FORM OF THE EARTH 

difference is not perceptible to the unaided eye will be 
apparent if the construction of such a figure is attempted, 
say ten inches in diameter in one direction and -^^ of an 
inch less in the opposite direction. The oblateness of 
Saturn is easily perceptible, being thirty times as great as 
that of the earth, or one tenth (see p. 257). Thus an 
ellipsoid nine inches in polar diameter (minor axis) and 
ten inches in equatorial diameter (major axis) would rep- 
resent the form of that planet. 

Although the oblateness of the earth seems slight when 
represented on a small scale and for most purposes may be 
ignored, it is nevertheless of vast importance in many 
problems in surveying, astronomy, and other subjects. 
Under the discussion of latitude it will be shown how this 
oblateness makes a difference in the lengths of degrees of 
latitude, and in the Appendix it is shown how this equa- 
torial bulging shortens the length of the year and changes 
the incUnation of the earth's axis (see Precession of the 
Equinoxes and Motions of the Earth's Axis). 

Dimensions of the Spheroid. It is of very great impor- 
tance in many ways that astronomers and surveyors know 
as exactly as possible the dimensions of the spheroid. 
Many men have made estimates based upon astronomical 
facts, pendulum experiments and careful surveys, as to the 
equatorial and polar diameters of the earth. Perhaps the 
most widely used is the one made by A. R. Clarke, for many 
years at the head of the English Ordnance Survey, known 
as the Clarke Spheroid of 1866. 

Clarke Spheroid op 1866. 

A. Equatorial diameter 7,926.614 miles 

B. Polar diameter 7,899.742 miles 

Oblateness — : — .... ^— 

A 295 



DIMENSIONS OF THE SPHEROID 31 

It is upon this spheroid of reference that all of the 
work of the United States Geological Survey and of the 
United States Coast and Geodetic Survey is based, and 
upon which most of the dimensions given in this book are 
determined. 

In 1878 Mr. Clarke made a recalculation, based upon 
additional information, and gave the following dimensions, 
though it is doubtful whether these approximations are 
any more nearly correct than those of 1866. 

Clarke Spheroid of 1878. 

A. Equatorial diameter 7,926.592 miles 

B. Polar diameter 7,899.580 miles 

Oblateness^^ ^^ 

Another standard spheroid of reference often referred 
to, and . one used by the United States Governmental 
Surveys before 1880, when the Clarke spheroid was 
adopted, was calculated by the distinguished Prussian 
astronomer, F. H. Bessel, and is called the 

Bessel Spheroid op 1841. 

A. Equatorial diameter 7,925.446 miles 

B. Polar diameter 7,898.954 miles 

Oblateness — -. — „■,„ ic 

A 299.16 

Many careful pendulum tests and a great amount of 
scientific triangulation surveys of long arcs of parallels 
and meridians within recent years have made available 
considerable data from which to determine the true 
dimensions of the spheroid. In 1900, the United States 
Coast and Geodetic Survey completed the measurement 
of an arc across the United States along the 39th parallel 



32 THE FOBM OF THE EARTH 

from Cape May, New Jersey, to Point Arena, California, 
through 48° 46' of longitude, or a distance of about 2,625 
miles. This is the most extensive piece of geodetic sur- 
veying ever undertaken by any nation, and was so carefully 
done that the total amount of probable error does not 
amount to more than about eighty-five feet. A long arc 
has been surveyed diagonally from Calais, Maine, to 
New Orleans, Louisiana, through 15° 1' of latitude and 
22° 47' of longitude, a distance of 1,623 miles. Another 
long arc will soon be completed along the 98th meridian 
across the United States. Many shorter arcs have also 
been surveyed in this country. 

The EngUsh government undertook in 1899 the gigantic 
task of measuring the arc of a meridian extending the 
entire length of Africa, from Cape Town to Alexandria. 
This will be, when completed, 65° long, about half on 
each side of the equator, and will be of great value in 
determining the oblateness. Russia and Sweden have 
lately completed the measurement of an arc of 4° 30' on 
the island of Spitzbergen, which from its high latitude, 
76° to 80° 30' N., makes it peculiarly valuable. Large 
arcs have been measured in India, Russia, France, and 
other countries, so that there are now available many times 
as much data from which the form and dimensions of the 
earth may be determined as Clarke or Bessel had. 

The late Mr. Charles A. Schott, of the United States 
Coast and Geodetic Survey, in discussing the survey of the 
39th parallel, with which he was closely identified, said:* 

" Abundant additional means for improving the existing 
deductions concerning the earth's figure are now at hand, 
and it is perhaps not too much to expect that the Interna- 

* In his Transcontinental Triangulation and the American Arc of 
the Parallel. 



EQUATOR ELLIPTICAL 33 

tional Geodetic Association may find it desirable in the 
near future to attempt a new combination of all available 
arc measures, especially since the two large arcs of the 
parallel, that between Ireland and Poland and that of the 
United States of America, cannot fail to have a paramount 
influence in a new general discussion." 

A spheroid is a solid nearly spherical. An oblate sphe- 
roid is one flattened toward the poles of its axis of rotation. 
The earth is commonly spoken of as a sphere. It would 
be more nearly correct to say it is an oblate spheroid. 
This, however, is not strictly accurate, as is shown in the 
succeeding discussion. 

The Earth a Geoid 

Conditions Producing Irregularities. If the earth had 
been made up of the same kinds of material imiformly 
distributed throughout its mass, it would probably have 
assumed, because of its rotation, the form of a regular 
oblate spheroid. But the earth has various materials 
unevenly distributed in it, and this has led to many slight 
variations from regularity in form. 

Equator Elliptical. Pendulum experiments and measure- 
ments indicate not only that meridians are elliptical but 
that the equator itself may be shghtly elliptical, its longest 
axis passing through the earth from 15° E. to 165° W. and 
its shortest axis from 105° E. to 75° W. The amount of this 
oblateness of the equator is estimated at about ^,^o„ or a 
difference of two miles in the lengths of these two diameters 
of the equator. Thus the meridian circle passing through 
central Africa and central Europe (15° E.) and around 
near Behring Strait (165° W.) may be shghtly more oblate 
than the other meridian circles, the one which is most 

JO. MATH. GEO. — 3 



34 



THE FORM OF THE EARTH 



nearly circular passing through central Asia (105° E.), 
eastern North America, and western South America 
(75° W.). 

United States Curved Unequally. It is interesting to 
note that the dimensions of the degrees of the long arc 
of the 39th parallel surveyed in the United States bear out 



ISO 120 80 40 



80 120 lUO 




100 120 80 iU 



80 120 100 



Fig. 13. Gravimetric lines showing variation in force of gravity 



to a remarkable extent the theory that the earth is slightly 
flattened longitudinally, making it even more than that 
just given, which was calculated by Sir John Herschel and 
A. R. Clarke. The average length of degrees of longitude 
from the Atlantic coast for the first 1,500 miles corresponds 
closely to the Clarke table, and thus those degrees are longer, 
and the rest of the arc corresponds closely to the Bessel 
table and shows shorter degrees. 



GEOID DEFINED 



36 



Cape May to Wallace (Kansas) 
Wallace to Uriah (CaUf.) . 



Diff. in 
long. 



26.661° 
21.618° 



Length 
of 1° 



53.829 mi. 
53.822 mi 



Clarke 



53.828 mi 



Bessel 



53.821 mi. 



Earth not an Ellipsoid of Three Unequal Axes. This 
oblateness of the meridians and oblateness of the equa- 
tor led some to treat the earth as an ellipsoid of three 
unequal axes: (1) the longest equatorial axis, (2) the 
shortest equatorial axis, and (3) the polar axis. It has 
been shown, however, that meridians are not true ellipses, 
for the amoimt of flattening northward is not quite the 
same as the amount southward, and the mathematical 
center of the earth is not exactly in the plane of the equator. 

Geoid Defined. The term geoid, which means " like the 
earth," is now applied to tliat mathematical figure which 
most nearly corresponds to the true shape of the Sarth. Moun- 
tains, valleys, and other slight deviations from evenness 
of surfaces are treated as departures from the geoid of 
reference. The following definition by Robert S. Wood- 
ward, President of the Carnegie Institution of Washington, 
very clearly explains what is meant by the geoid.* 

" Imagine the mean sea level, or the surface of the sea 
freed from the undulations due to winds and to tides. 
This mean sea surface, which may be conceived to extend 
through the continents, is called the geoid. It does not 
coincide exactly with the earth's spheroid, but is a slightly 
wavy surface lying partly above and partly below the 
spheroidal surface, by small but as yet not definitely known 
amounts. The determination of the geoid is now one of 
the most important problems of geophysics." 
* Encyclopaedia Americana. 



36 THE POEM OF THE EARTH 

An investigation is now in progress in the United States 
for determining a new geoid of reference upon a plan never 
followed hitherto. The following is a lucid description * 
of the plan by John F. Hayford, Inspector of Geodetic 
Work, United States Coast and Geodetic Survey. 

Area Method of Determining Form of the Earth. " The 
arc method of deducing the figure of the earth may ■ be 
illustrated by supposing that a skilled workman to whom is 
given several stiff wires, each representing a geodetic arc, 
either of a parallel or a meridian, each bent to the radius 
deduced from the astronomic observations of that arc, is 
told in what latitude each is located on the geoid and then 
requested to construct the ellipsoid of revolution which 
will conform most closely to the bent wires. Similarly, 
the area method is illustrated by supposing that the work- 
man is given a piece of sheet metal cut to the outUne of 
the continuous tiiangulation which is supplied with neces- 
sary astronomic observations, and accurately molded to fix 
the curvature of the geoid, as shown by the astronomic 
observations, and that the workman is then requested to 
construct the ellipsoid of revolution which will conform 
most accurately to the bent sheet. Such a bent sheet 
essentially includes within itself the bent wires referred to 
in the first illustration, and, moreover, the wires are now 
held rigidly in their proper relative positions. The sheet is 
much more, however, than this rigid system of bent hues, 
for each arc usually treated as a hne is really a belt of 
considerable width which is now utilized fully. It is obvi- 
ous that the workman would succeed much better in con- 
structing accurately the required ellipsoid of revolution 
from the one bent sheet than from the several bent wires. 
When this proposition is examined analytically it will be 
* Given at the International Geographic Congress, 1904. 



ON A MERIDIAN CIRCLE 37 

seen to be true to a much greater extent than appears 
from this crude illustration." 

" The area of irregular shape which is being treated as a 
single unit extends from Maine to California and from 
Lake Superior to the Gulf of Mexico. It covers a range of 
57° in longitude and 19° in latitude, and contains 477 
astronomic stations. This triangulation with its numerous 
accompan5dng astronomical observations will, even with- 
out combination with similar work in other countries, 
furnish a remarkably strong determination of the figure 
and size of the earth." 

It is possible that at some distant time in the future the 
dimensions and form of the geoid will be so accurately 
known that instead of using an oblate spheroid of reference 
(that is, a spheroid of such dimensions as most closely 
correspond to the earth, treated as an oblate spheroid such 
as the Clarke Spheroid of 1866), as is now done, it will be 
possible to treat any particular area of the earth as having 
its own peculiar curvature and dimensions. 

Conclusion. What is the form of the earth? We went 
to considerable pains to prove that the earth is a sphere. 
That may be said to be its general form, and in very many 
calculations it is always so treated. For more exact cal- 
culations, the earth's departures from a sphere must be 
borne in mind. The regular geometric solid which the 
earth most clearly resembles is an oblate spheroid. Strictly 
speaking, however, the form of the earth (not considering 
such irregularities as mountains and valleys) must be 
called a geoid. 

Directions on the Earth 

On a Meridian Circle. Think of yourself as standing 
on a great circle of the earth passing through the poles. 



88 THE FOKM OF THE EARTH 

Pointing from the northern horizon by way of your feet 
to the southern horizon, you have pointed to all parts 
of the meridian circle beneath you. Yoiir arm has 
swung through an angle of 180°, but you have pointed 
through all points of the meridian circle, or 360° of it. 
Drop your arm 90° or from the horizon to the nadir, 
and" you have pointed through half of the meridian 
circle, or 180° of latitude. It is apparent, then, that for 
every degree you drop your arm, you point through 
a space of two degrees of latitude upon the earth 
beneath. 

The north pole is, let us say, 45° from you. Drop 
your arm 22^° from the northern horizon, and you will 
point directly toward the north pole (Fig. 14). What- 
ever your latitude, drop your arm half as many degrees from 
the northern horizon as you are degrees from the pole, and you 
will point directly toward that pole* 

You may be so accustomed to thinking of the north 
pole as northward in a 

horizontal Une from you S. Horizon -J- N. Horizon 

that it does not seem ^^ V^^^^^^r;^^^^ 

real to think of it as x >^ — T 1I^^^k^'''°'^ 

below the horizon. This 

is because one is liable 

to forget that he is 

hving on a ball. To 

point to the horizon is Fig. m 

to point away from the earth. 

A Pointing Exercise. It may not be easy or even 
essential to learn exactly to locate many places in rela- 
tion to the home region, but the ability to locate readily 

* The angle included between a tangent and a chord is measured 
by one half the intercepted arc. 




A POINTING EXERCISE 



39 



Horizon Line 




Fig. 15 



some salient points greatly clarifies one's sense of loca- 
tion and conception of the earth as a baU. 

The following exer- 
cise is designed for 
students living not far 
from the 45th parallel. 
Since it is impossible 
to point the arm or 
pencil with accuracy 
at any given angle, it 
is roughly adapted for 
the north temperate 
latitudes (Fig. 15) . 
Persons living in the 
southern states may 
use Figure 16, based 
on the 30th parallel. 
The student should make the necessary readjustment for 

his own latitude. 

Horizon Line jy^^^ ^-^^ ^^^ j^^^ 

the northern horizon 
quarter way down, or 
22|° and you are 
pointing toward the 
north pole (Fig. 15). 
Drop it half way 
down, or 45° from the 
horizon, and you are 
pointing 45° the other 
side of the north pole, 
or half way to the 
equator, on the same parallel but on the opposite side of 
the earth, in opposite longitude. Were you to travel half 




Fig. 16 



40 THE FORM OF THE EARTH 

way around the earth in a due easterly or westerly direc- 
tion, you would be at that point. Drop the arm 22^° 
more, or 67^° from the horizon, and you are pointing 45° 
farther south or to the equator on the opposite side of the 
earth. Drop the arm 22^° more, or 90° from the horizon, 
toward your feet, and you are pointing toward our anti- 
podes, 45° south of the equator on the meridian opposite 
ours. Find where on the earth this point is. Is the 
familiar statement, " digging through the earth to China," 
based upon a correct idea of positions and directions on 
the earth? 

From the southern horizon drop the arm 22|^°, and you 
are pointing to a place having the same longitude but on 
the equator. Drop the arm 22|^° more, and you point to 
a place having the same longitude as ours but opposite 
latitude, being 45° south of the equator on our meridian. 
Drop the arm 22^° more, and you point toward the south 
pole. Practice until you can point directly toward any of 
these seven points without reference to the diagram. 

Latitude and Longitude 

Origin of Terms. Students often have difficulty in 
remembering whether it is latitude that is measured east 
and west, or longitude. When we recall the fact that 
to the people who first used these terms the earth was 
believed to be longer east and west than north and south, 
and now we know that owing to the oblateness of the 
earth this is actually the case, we can easily remember that 
longitude (from the Latin longus, long) is measured east 
and west. The word latitude is from the Latin latitudo, 
which is from latus, wide, and was originally used to 
designate measurement of the " width of the earth," or 
north and south. 



LONGITUDE 



41 



Antipodal Areas. From a globe one can readily ascer- 
tain the point which is exactly opposite any given one on 
the earth. The map showing antipodal areas indicates 
at a glance what portions of the earth are opposite each 
other; thus Australia Ues directly through the earth from 




Fig. 17. Map of Antipodal Areas 



mid-Atlantic, the point antipodal to Cape Horn is in 
central Asia, etc. 

Longitude is measured on parallels and is reckoned from 
some meridian selected as standard, called the prime 
meridian. The meridian which passes through the Royal 
Observatory at Greenwich, near London, has long been the 
prime meridian most used. In many countries the 
meridian passing through the capital is taken as the prime 
meridian. Thus, the Portuguese use the meridian of the 
Naval Observatory in the Royal Park at Lisbon, the 



42 THE FORM OF THE EARTH 

French that of the Paris Observatory, the Greeks that of 
the Athens Observatory, the Russians that of the Royal 
Observatory at Pulkowa, near St. Petersburg. 

In the maps of the United States the longitude is often 
reckoned both from Greenwich and Washington. The 
latter city being a trifle more than 77° west of Greenwich, 
a meridian numbered at the top of the map as 90° west 
from Greenwich, is numbered at the bottom as 13° west 
from Washington. Since the United States Naval Obser- 
vatory, the point in Washington reckoned from, is 77° 3' 
81" west from Greenwich, this is shghtly inaccurate. 
Among all English speaking people and in most nations of 
the world, unless otherwise designated, the longitude of a 
place is understood to be reckoned from Greenwich. 

The longitude of a place is the arc of the parallel inter- 
cepted between it and the prime meridian. Longitude 
may also be defined as the arc of the equator intercepted 
between the prime meridian and the meridian of the 
place whose longitude is sought. 

Since longitude is measured on parallels, and parallels 
grow smaller toward the poles, degrees of longitude are 
shorter toward the poles, being degrees of smaller circles. 

Latitude is measured on a meridian and is reckoned 
from the equator. The number of degrees in the arc of 
a meridian circle, from the place whose latitude is sought 
to the equator, is its latitude. Stated more formally, the 
latitude of a place is the arc of the meridian intercepted 
between the equator and that place. (See Latitude in 
Glossary.) What is the greatest number of degrees of 
latitude any place may have? What places have no 
latitude? 

Comparative Lengths of Degrees of Latitude. If the eartB" 
were a perfect sphere, meridian circles would be true mathe- 



LATITUDE 



43 



Pole 



matical circles. Since the earth is an oblate spheroid, 
meridian circles, so called, curve less rapidly toward the 
poles. Since the curvature is greatest near the equator, 
one would have to travel less distance on a meridian there 
to cover a degree of curvature, and a degree of latitude is 
thus shorter near the equator. Conversely, the meridian 
being slightly flat- 
tened toward the 
poles, one would 
travel farther there 
to cover a degree 
of latitude, hence 
degrees of latitude 
are longer toward 
the poles. Perhaps 
this may be seen 
more clearly from 
Figure 18. 

While all circles 
have 360° the de- 
grees of a small 
circle are, of course, 
shorter than the degrees of a greater circle. Now an 
arc of a meridian near the equator is obviously a part 
of a smaller circle than an arc taken near the poles and, 
consequently, the degrees are shorter. Near the poles, 
because of the flatness of a meridian there, an arc of a 
meridian is a part of a larger circle and the degrees are 
longer. As we travel northward, the North star (polestar) 
rises from the horizon. In traveling from the equator on 
a meridian, one would go 68.7 miles to see the polestar 
rise one degree, or, in other words, to cover one degree of 
curv9,ture of the meridian. Near the pole, where the earth 




Fig. i8 



44 



THE FORM OF THE EAETH 



is flattest, one would have to travel 69.4 miles to cover one 
degree of curvature of the meridian. The average length 
of a degree of latitude throughout the United States is 
almost exactly 69 miles. 

Table of Lengths of Degrees. The following table 
shows the length of each degree of the parallel and of the 
meridian at every degree of latitude. It is based upon 
the Clarke spheroid of 1866. 





Deg. 


Deg. 




Deg. 


Deg. 




Deg. 


Deg. 


Lat. 


Par. 


Mer. 


Lat. 


Par. 


Mer. 


Lat. 


Par. 


Mer. 




Miles 


Miles 




Miles 


Miles 




Miles 


Miles 


0° 


69.172 


68.704 


31° 


59.345 


68.890 


61° 


33.623 


69.241 


1 


69.162 


68.704 


32 


58.716 


68.901 


62 


32.560. 


69.251 


2 


69.130 


68.705 


33 


58.071 


68.912 


63 


31.488 


69.261 


3 


69.078 


68.706 


34 


57.407 


68.923 


64 


30.406 


69.271 


4 


69.005 


68.708 


36 


,56.725 


68.935 


65 


29.315 


69.281 


6 


68.911 


68.710 


36 


56.027 


68.946 


66 


28.215 


69.290 


6 


68.795 


68.712 


37 


55.311 


68.958 


67 


27'. 106 


69,299 


7 


68.660 


68.715 


38 


54.579 


68.969 


68 


25.988 


69.308 


8 


68.504 


68.718 


39 


53.829 


68.981 


69 


24.862 


69.316 


9 


68.326 


68.721 


40 


53.063 


68.993 


70 


23.729 


69.324 


10 


68.129 


68.725 


41 


52.281 


69.006 


71 


22.589 


69.332 


11 


67.910 


68.730 


42 


51.483 


69.018 


72 


21.441 


69.340 


12 


67.670 


68.734 


43 


50.669 


69.030 


73 


20.287 


69.347 


13 


67.410 


68.739 


44 


49.840 


69.042 


74 


19.127 


69.354 


14 


67.131 


68.744 


45 


48.995 


69.054 


76 


17.960 


69,360 


15 


66.830 


68.751 


46 


48.136 


69.066 


76 


16.788 


69.366 


16 


66.510 


68.757 


47 


47.261 


69.079 


77 


15.611 


69.372 


17 


66.169 


68.764 


48 


46.372 


69.091 


78 


14.428 


69.377 


18 


65.808 


68.771 


49 


45.469 


69.103 


79 


13.242 


69.382 


19 


65.427 


68.778 


60 


44.552 


69.115 


80 


12.051 


69.386 


20 


65.026 


68.786 


61 


43.621 


69.127 


81 


10.857 


69.390 


21 


64.606 


68.794 


52 


42.676 


69.139 


82 


9.659 


69.394 


22 


64.166 


68.802 


63 


41.719 


69.151 


83 


8.458 


69.397 


23 


63.706 


68.811 


54 


40.749 


69.163 


84 


7.255 


69.400 


24 


63 . 228 


68.820 


66 


39.766 


69.175 


85 


6.049 


69.402 


26 


62.729 


68.829 


56 


38.771 


69.186 


86 


4.842 


69.404 


26 


62.212 


68.839 


67 


37.764 


69.197 


87 


3.632 


69.405 


27 


61 . 676 


68.848 


68 


36.745 


69.209 


88 


2.422 


69.407 


28 


61.122 


68.858 


69 


35.716 


69.220 


89 


1.211 


69 , 407 


29 


60.548 


68,869 


60 


34.674 


69.230 


90 


0.000 


69.407 


30 


59.956 


68.879 















CHAPTER III 

the rotation of the earth 

The Celestial Sphere 

Apparent Dome of the Sky. On a clear night the stars 
twinkling all over the sky seem to be fixed in a dark dome 
fitting down around the horizon. This apparent concavity, 
studded with heavenly bodies, is called the celestial sphere. 
Where the horizon is free from obstructions, one can see 
half * of the celestial sphere at a given time from the same 
place. 

A line from one side of the horizon over the zenith point 
to the opposite side of the horizon is half of a great circle 
of the celestial sphere. The horizon line extended to the 
celestial sphere is a great circle. Owing to its immense 



If these lines met at a point 50,000 miles 
distant, the difference in their direction 
could not be measured. Such is the ratio 
of the diameter of the earth and the dis- 
tance to the very nearest of the stars. 



Fig. 19 




distance, a line from an observer at A (Fig. 19), pointing 
to a star will make a line apparently parallel to one from 
B to the same star. The most refined measurements at 

* No allowance is here made for the refraction of rays of light or the 
sUght curvature of the globe in the locality. 

45 



46 THE ROTATION OF THE EARTH 

present possible fail to show any angle whatever between 
them. 

We may note the following in reference to the celestial 
sphere. (1) The earth seems to be a mere point in the 
center of this immense hollow sphere. ('>) The stars, 
however distant, are apparently fixed in this sphere. 
(3) Any plane from the observer, if extended, will divide 
the celestial sphere into two equal parts. (4) Circles 
may be projected on this sphere and positions on it indi- 
cated by degrees in distance from estabhshed circles or 
points. 

Celestial Sphere seems to Rotate. The earth rotates on 
its axis (the term rotation applied to the earth refers to 
its daily or axial motion). To us, however, the earth 
seems stationary and the celestial sphere seems to rotate. 
Standing in the center of a room and turning one's body 
around, the objects in the room seem to rotate around in 
the opposite direction. The point overhead will be the 
only one that is stationary. Imagine a fly on a rotating 
sphere. If it were on one of the poles, that is, at the end 
of the axis of rotation, the object directly above it would 
constantly remain above it while every other fixed object 
would seem to swing around in circles. Were the fly to 
walk to the equator, the point directly away from the 
globe would cut the largest circle aroimd him and the 
stationary points would be along the horizon. 

Celestial Pole. The point in the celestial sphere directly 
above the pole and in line with the axis has no motion. 
It is called the celestial pole. The star nearest the pole 
of the celestial sphere and directly above the north pole 
of the earth is called the North star, and the star nearest 
the southern celestial pole the South star. It may be of 
interest to note that as we located the North star by refer- 



AT THE NORTH POLE 



47 




Fig. ao 



ence to the Big Dipper, the South star is located by refer- 
ence to a group of stars known as the Southern Cross. 

Celestial Equator. A great circle is conceived to extend 
around the celestial sphere 90° 
from the poles (Fig. 20). This 
is called the celestial equator. 
The axis of the earth, if pro- 
longed, would pierce the celes- 
tial poles, almost pierce the 
North and South stars, and 
the equator of the earth if ex- 
tended would coincide with the 
celestial equator. 

At the North Pole. An 
observer at the north pole 

will see the North star almost exactly overhead, and 
as the earth tm^ns around under his feet it will remain 
constantly overhead (Fig. 21). Halfway, or 90° from 
the North star, is the celestial equator around the 

horizon. As the earth 
rotates, — though it 
seems to us per- 
fectly still, — the stars 
around the sky seem 
to swing in circles in 
the opposite direction. 
The planes of the star 
paths are parallel to 
the horizon. The 
same half of the celestial sphere can be seen all of the 
time, and stars below the horizon always remain so. 

All stars south of the celestial equator being forever 
invisible at the north pole, Sirius, the brightest of the 




48 THE ROTATION OF THE EARTH 

stars, and many of the beautiful constellations, can never 
be seen from that place. How peculiar the view of the 
heavens must be from the pole, the Big Dipper, the 
Pleiades, the Square of Pegasus, and other star groups 
swinging eternally around in courses parallel to the 
horizon. When the sun, moon, and planets are in the 
portion of their courses north of the celestial equator, 
they, of course, will be seen throughout continued rotations 
of the earth until they pass below the celestial equator, 
when they will remain invisible again for long periods. 

The direction of the daily apparent rotation of the stars 
is from left to right (westward), the direction of the 
hands of a clock looked at from above. Lest the direction 
of rotation at the north pole be a matter of memory 
rather than of insight, we may notice that in the United 
States and Canada when we face southward we see the 
sun's daily course in the direction left to right (westward), 
and going poleward the direction remains the same though 
the sun approaches the horizon more and more as we 
approach the north pole. 

At the South Pole. An observer at the south pole, at 
the other end of the axis, will see the South star directly 
overhead, the celestial equator on the horizon, and the 
plane of the star circles parallel with the horizon. The 
direction of the apparent rotation of the celestial sphere 
is from right to left, counter-clockwise. If a star is seen 
at one's right on the horizon at six o'clock in the morning, 
at noon it >will be in front, at about six o'clock at night at 
his left, at midnight behind him, and at about six o'clock 
in the morning at his right again. 

At the Equator. An observer at the equator sees the 
stars in the celestial sphere to be very different in their 
positions in relation to himself. Remembering that he is 



BETWEEN EQUATOE AND POLES 



49 




standing with the Hne of his body at right angles to the 
axis of the earth, it is easy to understand why all the stars 
of the celestial sphere seem to be shifted around 90° from 
where they were at the poles. The celestial equator is a 
great circle extending 
from east to west 
directly overhead. 
The North star is seen 
on the northern hori- 
zon and the South 
star on the southern 
horizon. The planes 
of the circles followed 
by stars in their daily 
orbits cut the horizon 
at right angles, the horizon being parallel to the axis. At 
the equator one can see the entire celestial sphere, half at 
one time and the other half about twelve hours later. 

Between Equator 
and Poles. At places 
between the equator 
and the poles, the ob- 
server is liable to feel 
that a star rising due 
east ought to pass 
the zenith about six 
hours later instead 
of swinging slantingly 
around as it actually 
seems to do. This is because one forgets that the axis 
is not squarely under his feet excepting when at the 
equator. There, and there only, is the axis at right 
angles to the line of one's body when erect. The 

JO. MATH. SEO. — 4 




50 THE ROTATION OF THE EARTH 

apparent rotation of the celestial sphere is at right angles 
to the axis. 

Photographing the Celestial Sphere. Because of the 
earth's rotation, the entire celestial sphere seems to rotate. 
Thus we see stars daily circling around, the polestar 
always stationary. When stars are photographed, long 
exposures are necessary that their faint light may affect 
the sensitive plate of the camera, and the photographic 
instruments must be constructed so that they will move 
at the same rate and in the same direction as the stars, 
otherwise the stars will leave trails on the plate. When 
the photographic instrimient thus follows the stars in 
their courses, each is shown as a speck on the plate and 
comets, meteors, planets, or asteroids, moving at different 
rates and in different directions, show as traces. 

Rotation of Celestial Sphere is Only Apparent. For a long 
time it was beheved that the heavenly bodies rotated 
around the stationary earth as the center. It was only 
about five hundred years ago that the astronomer Coper- 
nicus established the fact that the motion of the sun and 
stars around the earth is only apparent, the earth rotating. 
We may be interested in some proofs that this is the case. 
It seems hard to beheve at first that this big earth, 25,000 
miles in circumference, can turn around once in a day. 
" Why, that would give us a whirhng motion of over a 
thousand miles an hour at the equator." " Who could 
stick to a merry-go-round going at the rate of a thousand 
miles an hour?" When we see, however, that the sun, 
93,000,000 miles away, would have to swing around in a 
course of over 580,000,000 miles per day, and the stars, at 
their tremendous distances, would have to move at unthink- 
able rates of speed, we see that it is far easier to believe 
that it is the earth and not the celestial sphere that rotates 



EASTWARD DEFLECTION OF FALLING BODIES 51 

daily. We know by direct observation that other planets, 
the sun and the moon, rotate upon their axes, and may 
reasonably infer that the earth does too. 

So far as the whirling motion at the equator is concerned, 
it does give bodies a shght tendency to fly off, but the 
amount of this force is only 2^9 as great as the attractive 
influence of the earth; that is, an object which would 
weigh 289 pounds at the equator, were the earth at rest, 
weighs a pound less because of the centrifugal force of 
rotation (see p. 14). 

Proofs of the Earth's Rotation 

Eastward Deflection of Falling Bodies. Perhaps the 
simplest proof of the rotation of the earth is one pointed 
out by Newton, although he had no means of demon- 
strating it. With his clear vision he said that if the earth 
rotates and an objejct were dropped from a considerable 
height, instead of falling directly toward the center of 
the earth in the direction of the plumb line,* it would be 
deflected toward the east'. Experiments have been made 
in the shafts of mines where air currents have been shut off 
and a slight but urmiistakable eastward tendency has been 
observed. 

During the summer of 1906, a number of newspapers 
and magazines in the United States gave accounts of the 
eastward falling of objects dropped in the deep mines of 
northern Michigan, one of which (Shaft No. 3 of the Tam- 
arack mine) is the deepest in the world, having a vertical 
depth of over one mile (and stiU digging!). It was stated 
that objects dropped into such a shaft never reached the 

* The slight geocentric deviations of the plumb line are explained 
on pp. 281-282. 



62 THE ROTATION OF THE EARTH 

bottom but always lodged among timbers on the east 
side. Some papers added a touch of the grewsome by 
implying that among the objects found clinging to the east 
side are " pieces of a dismembered human body " which 
were not permitted to fall to the bottom because of the rota- 
tion of the earth. Following is a portion of an account* 
by F. W. McNair, President of the Michigan College of 
Mines. 

McNair's Experiment.^ " Objects dropping into the shaft 
under ordinary conditions nearly always start with some 
horizontal velocity, indeed it is usually due to such initial 
velocity in the horizontal that they get into the shaft 
at all. Almost all common objects are irregular in shape, 
and, drop one of them ever so carefully, contact with the 
air through which it is passing soon deviates it from 
the vertical, giving it a horizontal velocity, and this when 
the air is quite still. The object slides one way or another 
on the air it compresses in front of it. Even if the body 
is a sphere, the air will cause it to deviate, if it is rotating 
about an axis out of the vertical. . Again, the air in the 
shaft is in ceaseless motion, and any obliquity of the 
currents would obviously deviate the falling body from 
the vertical, no matter what its shape. If the falling 
object is of steel, the magnetic influence of the air mains 
and steam mains which pass down the shaft, and which 
invariably become strongly magnetic, may cause it to 
swerve from a vertical course . . . 

" A steel sphere, chosen because it was the only con- 
venient object at hand, was suspended about one foot 
from the timbers near the western corner of the compart- 
ment. The compartment stands diagonally with refer- 
ence to the cardinal points. Forty-two hundred feet below 
* In the Mining and Scientific Press, July 14, 1906. 



McNAIR'S EXPERIMENT 53 

a clay bed was placed, having its eastern edge some five 
feet east of the point of suspension of the ball. When 
the ball appeared to be still the -suspending thread was 
burned, and the instant of the dropping of the ball was 
indicated by a prearranged signal transmitted by tele- 
phone to the observers below, who, watch in hand, waited 
for the sphere to strike the bed of clay. It failed to 
appear at all. Another like sphere was hung in the center 
of the compartment and the trial was repeated with the 
same result. The shaft had to be cleared and no more 
trials were feasible. Some months later, one of the spheres, 
presumably the latter one, was found by a timberman 
where it had lodged in the timbers 800 feet from the 
surface. 

" It is not probable, however, that these balls lodged 
because of the earth's rotation alone. . . . The matter is 
really more complicated than the foregoing discussion 
implies. It has received mathematical treatment from 
the great Gauss. According to his results, the deviation 
to the east for a fall of 5,000 feet at the Tamarack mine 
should be a Httle under three feet. Both spheres had that 
much to spare before striking the timbers. It is almost 
certain, therefore, that others of the causes mentioned in 
the beginning acted to prevent a vertical fall. At any 
rate, these trials serve to emphasize the urdikelihood that 
an object which falls into a deep vertical shaft, like those 
at the Tamarack mine, will reach the bottom, even when 
some care is taken in selecting it and also to start it verti- 
cally. 

" If the timbering permits lodgment, as is the case in 
most shafts, it may truthfully be said that if a shaft is 
deep in proportion to its cross section few indeed will be 
the objects falling into it which will reach the bottom, 



54 



THE ROTATION OF THE EARTH 



and such objects are more likely to lodge on the easterly 
side than on any other." 

The Foucault Experiment. Another simple demonstra- 
tion of the earth's rotation is by the celebrated Foucault 
experiment. In 1851, the French physicist, M. Leon Fou- 
cault, suspended from the dome of the Pantheon, in Paris, 
a heavy iron ball by wire two hundred feet long. A pin 

was fastened to the lowest 
side of the ball so that when 
swinging it traced a slight 
mark in a layer of sand placed 
beneath it. Carefully the 
long pendulum was set swing- 
ing. It was found that the 
path gradually moved around 
toward the right. Now either 
the pendulum changed its 
plane or the building was 
gradually turned around. By 
experimenting with a ball 
suspended from a ruler one 
can readily see that gradually 
turning the ruler will not 
change the plane of the 
swinging pendulum. If the 
pendulum swings back and forth in a north and south direc- 
tion, the ruler can be entirely turned aroimd. without chang- 
ing the direction of the pendulum's swing. If at the north 
pole a pendulum was set swinging toward a fixed star, say 
Arcturus, it would continue swinging toward the same 
star and the earth would thus be seen to turn aroimd in 
a day. The earth would not seem to turn but the pendu- 
lum would seem to deviate toward the right or clockwise. 




Fig. 24 



THE I-OUCAULT EXPERIMENT 



55 



Interesting Experiment in the Dome 
of the Pantheon. 

Veiu Tork Sun Spwsial Service 

Earls, Oct, 23.-TAn interestingr experi- 
ment under the auspices of thet astro- 
nomical society of France took. place yes- 
terday afternoon when ocular proof of the 
revolution . of. the earth was given by 



Conditions for Success. The Foucault experiment has 
been made in many places at different times. To be suc- 
cessful there should be a long slender wire, say forty feet 
or more in length, down the well of a stairway. The weight 
suspended should be heavy and spherical so that the 
impact against the air may not cause it to slide to one 
side, and there 

should be protec- SHOWING THE EARTH'S MOTION 
tion against drafts 
of air. A good sized 
circle, marked off 
in degrees, should 
be placed under it, 
with the center 
exactly under the 
baU when at rest. 
From the rate of 
the deviation the 
latitude may be 
easily determined 
or, knowing the 
latitude, the devia- 
tion ma;y be cal- 
culated. 

To Calculate Amount of Deviation. At first thought 
it might seem as though the floor would turn completely 
around imder the pendulum in a day, regardless of the 
latitude. It will be readily seen, however, that it is only 
at the pole that the earth would make one complete rota- 
tion under the pendulum in one day * or show a deviation 
of 15° in an hour. At the equator the pendulum will 
show no deviation, and at intermediate latitudes the rate 
* Strictly speaking, in one sidereal day. 



means of a pendulum, consistinB of a ball 
weiBhlns 60 pounds attached by a wire ,70 
yards In length to the Interior of the dome 
of the Pantheon. Mr. Chaumle, nilnlster 
of pvlblic instruction,- who presided, 
burned a string .that tied the weight to a 
pillar aria; the Immense pendulum began 
its journey.;',: Sand had been. I>lq,ce4 on the 
floor and each time the pendulum passed 
over it fl. new track was marked In regu- 
lar deviation, though the plane of the 
pendulum's swing remained uncbanged. 
The experiment was completely success- 
ful. -110 J. 

Fig. as 



56 THE ROTATION OF THE EARTH 

of deviation varies. Now the ratio of variation from the 
pole considered as one and the equator as zero is shown 
in the table of " natural sines " (p. 311). It can be 
demonstrated that the nimiber of degrees the plane of the 
pendulum will deviate in one hour at any latitude is found 
by multiplying 15° by the sine of the latitude. 

d = deviation 
^ = latitude 
.-. d = sine ^ X 15°. 

\Vhether or not the student has a very clear conception of 
what is meant by " the sine of the latitude " he may easily 
calculate the deviation or the latitude where such a pen- 
dulum experiment is made. 

Example. Suppose the latitude is 40°. Sine 40° = 
.6428. The hourly deviation at that latitude, then, is 
.6428 X 15° or 9.64°. Since the pendulum deviates 9.64° 
in one hour, for the entire circuit it will take as many 
hours as that number of degrees is contained in 360° or 
about 37^ hours. It is just as simple to calculate one's 
latitude if the amount of deviation for one hour is known. 
Suppose the plane of the pendulum is observed to deviate 
9° in an hour. 

Sine of the latitude X 15° = 9°. 
.-. Sine of the latitude = -fj, or .6000. 

From the table of sines we find that this sine, .6000, corre- 
sponds more nearly to that of 37° (.6018) than to the sine 
of any other whole degree, and hence 37° is the latitude 
where the hourly deviation is 9°. At that latitude it would 
take forty hours (360 -^ 9 = 40) for the pendulum to 
make the entire circuit. 



OTHER EVIDENCE 



57 



Table of Variations. The following table shows the 
deviation of the plane. of -the pendulum for one hour and 
the time required to make one entire rotation. 



Latitude 


Hourly 


Circuit of 


Latitude. 


Hourly- 


Circuit of 


Deviation. 


Pendulum. 


Deviation. 


Pendulum. 


5° 


1.31° 


275 hrs. 


50° 


11.49° 


31 hrs. 


10 


2.60 


138 


55 


12.28 


29 


15 


3.09 


117 


60 


12.99 


28 


20 


5.13 


70 


65 


13.59 


26 


25 


6.34 


57 


70 


14.09 


25.5 


30 


7.50 


48 


75 


14.48 


24.8 


35 


8.60 


42 


80 


14.77 


24.5 


40 


9.64 


37 


85 


14.94 


24.1 


45 


10.61 


■34 


90 


15.00 


24.0 



Other Evidence. Other positive evidence of the rotation 
of the earth we have in the fact that the equatorial winds 
north of the equator veer toward the east and polar winds 
toward the west — south of the equator exactly opposite — 
and this is precisely the result which would follow from the 
earth's rotation. Cyclonic winds in the northern hemi- 
sphere in going toward the area of low pressure veer toward 
the right and anti-cyclonic winds also veer toward the 
right in leaving areas of high pressure, and in the southern 
hemisphere their rotation is the opposite. No explanation 
of these well-known facts has been satisfactorily advanced 
other than the eastward rotation of the earth, which easily 
accounts for them. 

Perhaps the best of modern proofs of the rotation of 
the earth is demonstrated by means of the spectroscope. 
A discussion of this is reserved until the principles are 
explained (pp. 107, 108) in connection with the proofs of 
the earth's revolution. 



58 



THE ROTATION OF THE EARTH 



Velocity of Rotation 

The velocity of the rotation at the surface, in miles per 
hour, in different latitudes, is as follows: 



Latitude. 


Velocity. 


Latitude. 


Velocity. 


Latitude. 


Velocity. 





1038 


44 


748 


64 


456 


5 


1034 


45 


735 


66 


423 


10 


1022 


46 


722 


68 


390 


15 


1002 


47 


709 


70 


356 


20 


975 


48 


696 


72 


322 


25 


941 


49 


682 


74 


287 


30 


899 


50 


668 


76 


252 


32 


881 


51 


654 


78 


216 


34 


861 


52 


640 


80 


181 


-36 


840 


53 


626 


82 


145 


38 


819 


54 


611 


84 


109 


39 


807 


55 


596 


86 


73 


40 


796 


56 


582 


88 


36 


41 


784 


58 


551 


89 


18 


42 


772 


60 


520 


89J 


9 


43 


760 


62 


488 


90 






Uniform Rate of Rotation. There are theoretical grounds 
for believing that the rate of the earth's rotation is getting 
gradually slower. As yet, however, not the slightest 
variation has been discovered. Before attacking the 
somewhat complex problem of time, the student should 
clearly bear in mind the fact that the earth rotates on its 
axis with such unerring regularity that this is the most 
perfect standard for any time calculations known to 
science. 

Determination of Latitude 

Altitude of Celestial Pole Equals Latitude. Let us return, 
in imagination, to the equator. Here we may see the North 
star on the horizon due north of us, the South star on the 



TO FIND TOUR LATITUDE 59 

horizon due south, and halfway between these two points, 
extending from due east through the zenith to due west, 
is the celestial equator. If we travel northward we shall 
be able to see objects which were heretofore hidden from 
view by the curvature of the earth. We shall find that 
the South star becomes hidden from sight for the same 
reason and the North star seems to rise in the sky. The 
celestial equator no longer extends through the point 
directly overhead but is somewhat to the south of the 
zenith, though it still intersects the horizon at the east 
and west points. As we go farther north this rising of 
the northern sky and sinking of the southern sky becomes 
greater. If we go halfway to the north pole we shall find 
the North star halfway between the zenith and the northern 
horizon, or at an altitude of 45° above the horizon. For 
every degree of curvature of the earth we pass over, 
going northward, the North star rises one degree from 
the horizon. At New Orleans the North star is 30° from 
the horizon, for the city is 30° from the equator. At 
Philadelphia, 40° north latitude, the North star is 40° 
from the horizon. South of the equator the converse of 
this is true. The North star sinks from the horizon and 
the South star rises as one travels southward from the 
equator. The altitude of the North star is the latitude of 
a place north of the equator and the altitude of the South 
star is the latitude of a place south of the equMtor. It is 
obvious, then, that the problem of determining latitude is 
the problem of determining the altitude of the celestial 
pole. 

To Find Your Latitude. By means of the compasses 
and scale, ascertain the altitude of the North star. This 
can be done by placing one side of the compasses on 
a level window sill and sighting the other side toward 



60 THE ROTATION OF THE EARTH 

the North star, then measuring the angle thus formed. 
Another simple process for ascertaining latitude is to 
determine the altitude of a star not far from the North 
star when it is highest and when it is lowest; the average 
of these altitudes is the altitude of the pole, or the latitude. 
This may easily be done in latitudes north of 38° during 
the winter, observing, say, at 6 o'clock in the morning 
and at 6 o'clock in the evening. This is simple in that 
it requires no tables. Of course, such measurements are 
very crude with simple instruments, but with a Uttle 
care one wiU usually be surprised at the accuracy of his 
results. 

Owing to the fact that the North star is not exactly at 
the north pole of the celestial sphere, it has a slight rotary 
motion. It will be more accurate, therefore, if the obser- 
vation is made when the Big Dipper and Cassiopeia are 



A 



True Pole 



True Pole 

;®« 



C Q True Pole ^ „ , a r, 

TruePoleQD 



Fig. 26 

in one of the positions (A or B) represented by Figure 26. 
In case of these positions the altitude of the North star 
will give the true latitude, it then being the same altitude 
as the pole of the celestial sphere. In case of position 
D, the North star is about 1{° below the true pole, hence 



TO PIND YOUR LATITUDE 61 

1^° must be added to the altitude of the star. In case of 
position C, the North star is 1^° above the true pole, and 
that amount must be subtracted from its altitude. It is 
obvious from the diagrams that a true north and south 
line can be struck when the stars are in positions C and D, 
by hanging two plumb Unes so that they he in the same 
plane as the zenith meridian Une through Mizar and Delta 
Cassiopeia. Methods of determining latitude will be further 
discussed on pp. 172-174. The instrument commonly 
used in observations for determining latitude is the meri- 
dian circle, or, on shipboard, the sextant. Read the de- 
scription of these instnmaents in any text on astronomy. 

Queries 

In looking at the heavenly bodies at night do the stars, 
moon, and planets all look as though they were equally 
distant, or do some appear nearer than others? The fact 
that people of ancient times beheved the celestial sphere 
to be made of metal and all the heavenly bodies fixed or 
moving therein, would indicate that to the observer who is 
not biased by preconceptions, all seem equally distant. 
If they did not seem equally distant they would not 
assume the apparently spherical arrangement. 

The decUnation, or distance from the celestial equator, 
of the star (Benetnasch) at the end of the handle of the 
Big Dipper is 40°. How far is it from the celestial pole? 
At what latitude will it touch the horizon in its swing 
under the North star? How far south of the equator could 
one travel and still see that star at some time? 



CHAPTER IV 

longitude and time 
Solar Time 

Stin Time Varies. The sun is the world's great time- 
keeper. He is, however, a slightly erratic one. At the 
equator the length of day equals the length of night the 
year through. At the poles there are six months day and 
six months night, and at intermediate latitudes the time 
of sunrise and of sunset varies with the season. Not only 
does the time of sunrise vary, but the time it takes the sun 
apparently to swing once around the earth also varies. 
Thus from noon by the sun until noon by the sun again is 
sometimes more than twenty-four hours and sometimes 
less than twenty-foior hours. The reasons for this varia- 
tion will be taken up in the chapter on the earth's revolution. 

Mean Solar Day. By a mean solar day is meant the 
average interval of time from sun noon to sun noon. 
While the apparent solar day varies, the mean solar day 
is exactly twenty-four hours long. A sundial does not 
record the same time as a clock, as a usual thing, for the 
sundial records apparent solar time while the clock records 
mean solar time. 

Relation of Longitude to Time. The sun's apparent daily 
journey around the earth with the other bodies of tfie 
celestial sphere gives rise to day and night.* It takes the 
sun, on the average, twenty-four hours apparently -to swing 

* Many thoughtlessly assume that the fact of day and night is a 
proof of the earth's rotation. 

62 



HOW LONGITUDE IS DETERMINED 63 

once around the earth. In this daily journey it crosses 
360° of longitude, or 15° for each hour. It thus takes 
four minutes for the sun's rays to sweep over one degree 
of longitude. Suppose it is noon by the sun at the 90th 
meridian, in four minutes the sun will be over the 91st 
meridian, in four more minutes it will be noon by the sun 
on the .92d meridian, and so on around the globe. 

Students are sometimes confused as to the time of day 
in places east of a given meridian as compared with the 
time in places west of it. When the sun is rising here, it 
has already risen for places east of us, hence their time is 
after sunrise or later than ours. If it is noon by the 
sun here, at places east of us, having already been noon 
there, it must be past noon or later in the day. Places 
to the east hare later time because the sun reaches them first. 
To the westward the converse of this is true. If the sun 
is rising here, it has not yet risen for places west of us and 
their time is before sunrise or earher. When it is noon by 
the sun in Chicago, the shadow north, it is past noon by 
the sun in Detroit and other places eastward and before 
noon by the sun in Minneapolis and other places westward. 

How Longitude is Determined. A man when in London, 
longitude 0°, set his watch according to mean solar time 
there. When he arrived at home he found the mean solar 
time to be six hours earlier (or slower) than his watch, 
which he had not changed. Since his watch indicated 
later time, London must be east of his home, and since the 
sun appeared six hours earlier at London, his home must 
be 6 X 15°, or 90°, west of London. While on shipboard 
at a certain place he noticed that the sun's shadow was due 
north when his watch indicated 2 : 30 o'clock, p.m. Assum- 
ing that both the watch and the sun were " on time " we 
readily see that since London time was two and one half 



64 LONGITUDE AND TIME 

hours later than the time at that place, he must have been 
west of London 2^ X 15°, or 37° 30'. 

Ship's Chronometer. Every ocean vessel carries a very 
accurate watch called a chronometer. This is regulated 
to run as perfectly as possible and is set according to the 
mean solar time of some well known meridian. Vessels 
from Enghsh speaking nations all have their chronometers 
set with Greenwich time. By observing the time accord- 
ing to the sun at the place whose longitude is sought and 
comparing that time with the time of the prime meridian 
as indicated by the chronometer, the longitude is reck- 
oned. For example, suppose the time according to the 
sun is found by observation to be 9 : 30 o'clock, a.m., and 
the chronometer indicates 11:20 o'clock, a.m. The prime 
meridian, then, must be east as it has later time. Since 
the difference in time is one hour and fifty minutes and 
there are 15° difference in longitude for an hour's differ- 
ence in time, the difference in longitude must be If X 15°, 
or 27° 30'. 

The relation of longitude and time should be thoroughly 
mastered. From the table at the close of this chapter, 
giving the longitude of the principal cities of the world, 
one can determine the time it is in those places when it is 
noon at home. Many other problems may also be sug- 
gested. It should be borne in mind that it is the mean 
solar time that is thus considered, which in most cities is 
not the time indicated by the watches and clocks there. 
People all over Great Britain set their timepieces to agree 
with Greenwich time, in Ireland with DubUn, in France 
with Paris, etc. (see " Time used in Various Countries " at 
the end of this chapter). 

Local Time. The mean solar time of any place is often 
called its local time. This is the average time indicated 



ORIGIN OF PRESENT SYSTEM 65 

by the sundial. All places on the same meridian have the 
same local time. Places on different meridians must of 
necessity have different local time, the difference in time 
being four minutes for every degree's difference in longituder 

Standard Time 

Origin of Present System. Before the year 1883, the 
people of different cities in the United States commonly 
used the local time of the meridian passing through the city. 
Prior to the advent of the raihoad, telegraph, and telephone, 
little inconvenience was occasioned by the prevalence of so 
many time systems. But as transportation and communi- 
cation became rapid and complex it became very difficult to 
adjust one's time and calculations according to so many 
standards as came to prevail. Each railroad had its own 
arbitrary system of time, and where there were several 
railroads in a city there were usually as many species of 
" railroad time " besides the local time according to 
longitude. 

" Before the adoption of standard time there were some- 
times as many as five different kinds of time in use in a 
single town. The railroads of the United States followed 
fifty-three different standards, whereas they now use five. 
The times were very much out of joint." * 

His inability to make some meteorological calculations 
in 1874 because of the diverse and doubtful character of 
the time of the available weather reports, led Professor 
Cleveland Abbe, for so many years connected with the 
United States Weather service, to suggest that a system of 
standard time should be adopted. At about the same time 
several others made similar suggestions and the subject was 
soon taken up in an official way by the railroads of the 
* The Scrap Book, May, 1906. 

JO. MATH. GEO.— 5 



66 



LONGITUDE AND TIME 




MOUNTAIN STANDARD TIME 67 

country under the leadership of William F. Allen, then sec- 
retary of the General Time Convention of Railroad Officials. 
As a result of his untiring efforts the railway associations 
endorsed his plan and at noon of Sunday, November 18, 
1883, the present system was inaugurated. 

Eastern Standard Time. Accordiag to the system all 
cities approximately within 7-}° of the 75th meridian use 
the mean solar time of that meridian, the clocks and 
watches being thus just five hours earlier than those of 
Greenwich. This belt, about 15° wide, is called the eastern 
standard time belt. The 75th meridian passes through the 
eastern portion of Philadelphia, so the time used through- 
out the eastern portion of the United States corresponds 
to Philadelphia local mean solar time. 

Central Standard Time. The time of the next belt is the 
mean solar time of the 90th meridian or one hour slower 
than eastern standard time. This meridian passes through 
or very near Madison, Wisconsin, St. Louis, and New 
Orleans, where mean local time is the same as standard 
time. When it is noon at Washington, D. C, it is 11 
o'clock, A.M., at Chicago, because the people of the former 
city use eastern standard time and those at the latter use 
central standard time. 

Mountain Standard Time. To the west of the central 
standard time belt hes the mountain region where the 
time used is the mean solar time of the 105th meridian. 
This meridian passes through Denver, Colorado, and its 
clocks as a consequence indicate the same time that the 
mean sim does there. As the standard time map shows, 
all the belts are bounded by irregular fines, due to the 
fact that the people of a city usually use the same time 
that their principal railroads do, and where trains change 
their time depends in a large measure upon the conven- 



68 LONGITUDE AND TIME 

ience to be served. This belt shows the anomaly of being 
botmded on the east by the central time belt, on the west 
by the Pacific time belt, and on the south by the same belts. 
The reasons why the mountain standard time belt tapers 
to a point at the south and the peculiar conditions which 
consequently result, are discussed under the topic " Four 
Kinds of Time around El Paso " (p. 75). 

Pacific Standard Time. People living in the states bor- 
dering or near the Pacific Ocean use the mean solar time 
of the 120th meridian and thus have three hours earlier 
time than the people of the Atlantic coast states. This 
meridian forms a portion of the eastern boundary of 
Cahfornia. 

In these great time belts * all the clocks and other time- 
pieces differ in time by whole hours. In addition to astron- 
omical observatory clocks, which are regulated according 
to the mean local time of the meridian passing through the 
observatory, there are a few cities in Michigan, Georgia, 
New Mexico, and elsewhere in the United States, . where 
mean local time is still used. 

Standard Time in Europe. In many European countries 
standard time based upon Greenwich time, or whole hour 
changes from it, is in general use, although there are many 
more cities which use mean local time than in the United 
States. Western European time, or that of the meridian 
of Greenwich, is used in Great Britain, Spain, Belgium, and 
Holland. Central European time, one hour later than that 
of Greenwich, is used in Norway, Sweden, Denmark, Luxem- 
burg, Germany, Switzerland, Austria-Hungary, Servia, and 
Italy. Eastern European time, two hours later than that 
of Greenwich, is used in Turkey, Bulgaria, and Roumania. 

* For a discussion of the time used in other portions of North 
America and elsewhere in the world see pp. 81-87. 



GETTING THE TIME 69 



Telegraphic Time Signals 

Getting the Time. An admirable system of sending time 
signals all over the country and even to Alaska, Cuba, and 
Panama, is in vogue in the United States, having been 
established in August, 1865. The Naval Observatories at 
Washington, D. C, and Mare Island, California, send out 
the signals during the five minutes preceding noon each 
day. 

It is a common custom for astronomical observatories to 
correct their own clocks by careful observations of the stars. 
The Washington Observatory sends telegraphic signals to 
all the cities east of the Rocky Moimtains and the Mare 
Island Observatory to Pacific cities and Alaska. A few 
raihoads receive their time corrections from other observ- 
atories. Goodsell Observatory, Carleton College, Ndrth- 
field, Minnesota, has for many years furnished time to the 
Great Northern, the Northern Pacific, the Great Western, 
and the Sault Ste. Marie railway systems. Allegheny 
Observatory sends out time to the Pennsylvania system 
and the Lick Observatory to the Southern Pacific system. 

How Time is Determined at the United States Naval 
Observatory. The general plan of correcting clocks at the 
United States Naval Observatories by steUar observations 
is as follows: A telescope called a meridian transit is situ- 
ated in a true north-south direction mounted on an east- 
west a,xis so that it can be rotated in the plane of the 
meridian but not in the slightest degree to the east or 
west. Other instrmnents Used are the chronograph and 
the sidereal clock. The chronograph is an instrument 
which may be electrically connected with the clock and 
which automatically makes a mark for each second on a 
sheet of paper fastened to a cylinder. The sidereal clock 



70 LONGITUDE AND TIME 

is regulated to keep time with the stars — not with the 
sun, as are other clocks. The reason for this is because 
the stars make an apparent circuit with each rotation of 
the earth and this, we have observed, is unerring while 
the sun's apparent motion is qiiite irregular. 

To correct the clock, an equatorial or high zenith star 
is selected. A well known one is chosen since the exact 
time it will cross the meridian of the observer (that is, be 
at its highest point in its apparent daily rotation) must be 
calculated. The chronograph is then started, its pen and 
ink adjusted, and its electrical wires connected with the 
clock. The observer now sights the telescope to the point 
where the expected star will cross his meridian and, with 
his hand on the key, he awaits the appearance of the star. 
As the star crosses each of the eleven hair lines in the field 
of the telescope, the observer presses the key which auto- 
matically marks upon the chronographic cylinder. Then 
by examining the sheet he can tell at what time, by the 
clock, the star crossed the center line. He then calculates 
just what time the clock should indicate and the difference 
is the error of the clock. By this means an error of one 
tenth of a second can be discovered. 

Jhe Sidereal Clock. The following facts concerning the 
sidereal clock may be of interest. It is marked with 
twenty-four hour spaces instead of twelve. Only one 
moment in the year does it indicate the same time as 
ordinary timepieces, which are adjusted to the average sun. 
When the error of the clock is discovered the clock is 
not at once reset because any tampering with the clock 
would involve a slight error. The correction is simply 
noted and the rate of the clock's gaining or losing time is 
calculated, so that the true time can be ascertained very 
precisely at any time by referring to the data showing the 



SENDING TIME SIGNALS 71 

clock error when last corrected and the rate at which it 
varies. 

A while before noon each day the exact sidereal time is 
calculated ; this is converted into local mean solar time and 
this into standard time. The Washington Naval Observ- 
atory converts this into the standard time of the 75th 
meridian or Eastern time and the Mare Island Observatory 
into that of the 120th meridian or Pacific time. 

Sending Time Signals. By the cooperation of telegraph 
companies, the time signals which are sent out daily from 
the United States Naval Observatories reach practically 
every telegraph station in the country. They are sent 
out at noon, 75th meridian time, from ^Yashington, which 
is 11 o'clock, A.M., in cities using Central time and 10 
o'clock, A.M., where Mountain time prevails; and at noon, 
120th meridian time, they are sent' to Pacific coast cities 
from the Mare Island Observatory — ■ three hours after 
Washington has flashed the signal which makes correct 
time accessible to sixty millions of our population living 
east of the Rockies. 

Not only are the time signals sent to the telegraph sta- 
tions and thence to railway ofl!ices, clock makers and 
repairers, schools, court houses, etc., but the same tele- 
graphic signal that marks noon also actually sets many 
thousands of clocks, their hands whether fast or slow auto- 
matically flying to the true mark in response to the electric 
current. In a number of cities of the United States, 
nineteen at present, huge baUs are placed upon towers or 
buildings and are automaticaUy dropped by the electric 
noon signal. The time ball in Washington is conspicu- 
ously placed on the top of the State, War, and Navy build- 
ing and may be seen at considerable distances from many 
parts of the city. 



72 



LONGITUDE AND TIME 



A few minutes before noon each day, one wire at each 
telegraphic office is cleared of all business and " thousands 
of telegraph operators sit in silence, waiting for the cHck 
of the key which shall tell them that the ' master clock ' 
in Washington has begun to speak." * At five minutes 
before twelve the instrument begins to chck off the seconds. 
Figure 28 (adapted from a cut appearing in Vol. IV, Appen- 
dix IV, United States Naval Observatory Publications) 



10 



20 



30 



40 



so 



60 



L^^^-,.^..^ ,1 . . 1 1 , ■ ■ 1 1 ... 1 

55*'' Minute before noon SSecomitted 


56"' Minute-before noon 5 Sec omitted 
1 1 1 1 1 1 


57"' Minute before noon SSecomitted 
1 1 1 1 1.... 1 


SB"" Minute before noon SSecomitted 
.... 1 1 1 1 1 Noon 1 



59 *'! Minute before noon 
Fig. 28 



10 Sec omitted 



graphically shows which second beats are sent along the 
wires during each of the five minutes before noon by the 
transmitting clock at the Naval Observatory. 

Explanation of the Second Beats. It will be noticed 
that the twenty-ninth second of each minute is omitted. 
This is for the purpose of permitting the observer to 
distinguish, without counting the beats, which is the one 
denoting the middle of each minute; the five seconds at 
the end of each of the first four minutes are omitted to 
mark the beginning of a new minute and the last ten 
seconds of the fifty-ninth minute are omitted to mark 
conspicuously the moment of noon. The omission of the 

* From "What's the Time," Youth's Companion, May 17 and 
June 14, 1906. 



SENDING TIME SIGNALS 73 

last ten seconds also enables the operator to connect 
his wire with the clock to be automatically set or the 
time ball to be dropped. The contact marking noon is 
prolonged a full second, not only to make prominent this 
important moment but also to afford sufficient current 
to do the other work which this electric contact must 
perform. 

Long Distance Signals. Several times in recent years 
special telegraphic signals have been sent' out to such dis- 
tant points as Madras, Mauritius, Cape Town, Pulkowa 
(near St. Petersburg), Rome, Lisbon, Madrid, Sitka, 
Buenos Ayres, Wellington, Sydney, and Guam. Upon 
these occasions " our standard clock may fairly be said to 
be heard in ' the remotest ends of the earth,' thus antici- 
pating the day when wireless telegraphy will perhaps allow 
of a daily international time signal that will reach every 
contineiit and ocean in a small fraction of a second. "* 

These reports have been received at widely separated 
stations within a few seconds, being received at the Lick 
Observatory in 0.05*, Manila in 0.11°, Greenwich in 
1.33^ and Sydney, Australia, in 2.25°. 

Confusion from Various Standards 

Where different time systems are used in the same 
community, confusion must of necessity result. The 
following editorial comment in the Official Railway Ouide 
for November, 1900, very succinctly sets forth the con- 
dition which prevailed in Detroit as regards standard and 
local time. 

" The city of Detroit is now passing through an agitation 
wliich is a reminiscence of those which took place through- 

* "The Present Status of the Use of Standard Time." by Lieut. 
Commander E. E. Hayden, U. S. Navy. 



74 LONGITUDE AND TIME 

out the country about seventeen years ago, when stand- 
ard time was first adopted. For some reason, which it 
is difficult to explain, the city fathers of Detroit have 
refused to change from the old local time to the standard, 
notwithstanding the fact that all of the neighboring cities 
— Cleveland, Toledo, Columbus, Cincinnati, etc., — in 
practically the same longitude, had made the change years 
ago and realized the benefits of so doing. The business 
men of Detroit and visitors to that city have been for a 
long time laboring under many disadvantages owing to 
the confusion of standards, and they have at last taken 
the matter into their own hands and a lively campaign, 
with the cooperation of the newspapers, has been 
organized during the past two months. Many of the 
hotels have adopted standard time, regardless of the 
city, and the authorities of Wayne County, in which 
Detroit is situated, have also decided to hold court on 
Central Standard time, as that is the official standard of 
the state of Michigan. The authorities of the city have^ 
so far not taken action. It is announced in the news- 
papers that they probably will do so after the election, 
and by that time, if progress continues to be made, the 
only clock in town keeping the local time will be on the 
town hall. All other matters will be regulated by standard 
time, and the hours of work will have been altered 
accordingly in factories, stores, and schools. Some 
opposition has been encountered, but this, as has been 
the case in every city where the change has been made; 
comes frorn people who evidently do not, comprehend the 
effects of the change. One individual, for instance, writes 
to a newspaper that he will decline to pay pew rent in 
any church whose clock tower shows standard time; he 
refuses to have his hours of rest curtailed. How these will 



roUR KINDS OF TIME ABOtTND EL PASO 



75 



be affected by the change he does not explain. Every 
visitor to Detroit who has encountered the difficulties 
which the confusion of standards there gives rise to, will 
rejoice when the complete change is effected." 

The longitude of Detroit being 83° W., it is seven degrees 
east of the 90th meridian, hence the local time used in the 
city was twenty-eight minutes faster than Central time 
and thirty-two minutes slower than Eastern time. In 
Gainesville, Georgia, mean local sun time is used in the city, 




Fig. 29 

while the Southern railway passing through the city uses 
Eastern time and the Georgia railway uses Central time. 

Four Kinds of Time Around El Paso. Another place of 
peculiar interest in connection with this subject is El Paso, 
Texas, from the fact that four different systems are em- 
ployed. The city, the Atchison, Topeka, and Santa Fe, 
and the El Paso and Southwestern railways use Mountain 
time. The Galveston, Harrisburg, and San Antonio, and 
the Texas and Pacific railways use Central time. The 
Southern Pacific railway uses Pacific time. The Mexican 
Central railway uses Mexican standard time. 



76 LOl^GlTUDE AND TIME 

From this it will be seen that when clocks in Strauss, 
N. M., a few miles from El Paso, are striking twelve, the 
clocks in El Paso are striking one; in Ysleta, a few miles 
east, they are strildng two; while across the river in Juarez, 
Mexico, the clocks indicate 12:24. 

Time Confusion for Travelers. The confusion which 
prevails where several different standards of time obtain 
is well illustrated in the following extract from " The 
Impressions of a Careless Traveler " by Ljmaan Abbott, 
in the Outlook, Feb. 28, 1903. 

" The changes in time are almost as interesting and 
quite as perplexing as the changes in currency. Of course 
our steamer time changes every day; a sharp blast on the 
whistle notifies us when it is twelve o'clock, and certain 
of the passengers set their watches accordingly every 
day. I have too much respect for my faithful friend to 
meddle mth him to this extent, and I keep my watch 
imchanged and make my calculations by a mental com- 
parison of my watch with the ship's time. But when we 
are in port we generally have three times — ship's time, 
local time, and railroad time, to which I must in my own 
case add my own time, which is quite frequently neither. 
In fact, I kept New York time till we reached Genoa; 
since then I have kept central Europe railroad time. 
Without changing my watch, I am getting back to that 
standard again, and expect to find myself quite accurate 
when we land in Naples." 

The Legal Aspect op Standard Time 

The legal aspect of standard time presents many 
interesting features. Laws have been enacted in many 
different countries and several of the states of this country 
legalizing some standard of time. Thus in Michigan, 



THE LEGAL ASPECT OF STANDARD TIME 77 

Minnesota, and other central states the legal time is the 
mean solar time of longitude 90° west of Greenwich. 
When no other standard is explicitly referred to, the 
time of the central belt is the legal time in force. Similarly, 
legal time in Germany was declared by an imperial decree 
dated March 12, 1903, as follows: * 

"We, Wilhelm, by the grace of God German Emperor, King of 
Prussia, decree in the name of the Empire, the Bundesrath and Reich- 
stag concurring, as follows : 

" The legal time in Germany is the mean solar time of longitude 15° 
east from Greenwich." 

Greenwich time for Great Britain, and Dublin time for 
Ireland, were legahzed by an act of Parhament as follows: 

A Bill to remove doubts as to the meaning of expressions relative 
to time occurring in acts of Parliament, deeds, and other legal instru- 
ments. 

Whereas it is expedient to remove certain doubts as to whether 
expressions of time occurring in acts of Parliament, deeds, and other 
legal instruments relate in England and Scotland to Greenwich time, 
and in Ireland to Dublin time, or to the mean astronomical time in 
each locality: 

Be it therefore enacted by the Queen's most Excellent Majesty, 
by and with the advice and consent of the Lords, spiritual and tem- 
poral, arid Commons, in the present Parliament assembled, and by 
the authority of the same, as follows (that is to say): 

1. That whenever any expression of time occurs in any act, of 
Parliament,, deed, or other legal instrument, the time referred to 
shall, unless it is otherwise specifically stated, be held in the case of 
Great Britain to be Greenwich mean time and in the case of Ireland, 
Dublin mean time. 

2. This act may be cited as the statutes (definition of time) act, 1880. 

Seventy-fifth meridian time was legahzed in the District 
of Columbia by the following act of Congress: 

An Act to establish a standard of time in the District of Columbia. 
Be it enacted by the Senate and House of Representatives of the 

* Several of the following quotations are taken from the " Present 
Status of the Use of Standard Time," by E. E. Hayden. 



78 LONGITUDE AND TIME 

United States of America in Congress assembled, That the legal 
standard of time in the District of Columbia shall hereafter be the 
mean time of the seventy-fifth meridian of longitude west from 
Greenwich. 

Section 2. That this act shall not be so construed as to affect 
existing contracts. 

Approved, March 13, 1884. 

In New York eastern standard time is legalized in 
section 28 of the Statutory Construction Law as follows: 

The standard time throughout this State is that of the 7Stli meridian 
of longitude west from Greenwich, and all courts and public offices, and 
legal and official proceedings, shall be regulated thereby. Any act 
required by or in pursuance of law to be performed at or within a pre- 
scribed time, shall be performed according to such standard time. 

A New Jersey statute provides that the time of the 
same meridian shall be that recognized in aU the courts and 
pubUc offices of the State, and also that " the time named 
in any notice, advertisement, or contract shall be deemed 
and taken to be the said standard time, unless it be other- 
wise expressed." In Pennsylvania also it is provided 
that " on and after July 1, 1887, the mean solar time of 
the seventy-fifth meridian of longitude west of Greenwich, 
commonly called eastern standard time," shall be the 
standard in all pubhc matters; it is further provided that 
the time " in any and all contracts, deeds, wills, and 
notices, and in the transaction of all matters of business, 
public, legal, commercial, or otherwise, shall be construed 
with reference to and in accordance with the said standard 
hereby adopted, unless a different standard is therein 
expressly provided for." 

Where there is no standard adopted by legal authority, 
difficulties may arise, as the following clipping from the 
New York Sun, November 25, 1902, illustrates: 



THE LEGAL ASPECT OF STANDARD TIME 



79 



WHAT'S NOON IN A FIRE 
POLICY ? 



Solar Noon or Standard Time 
Noon — Courts Asked to Say. 



Fire in Louisville at 11: 45 a.m.. Stand- 
ard Time, Which Was 12: 02 1-2 p.m. 
Solar Time — Policies Expired at 
Noon and 13 Insurance Companies 
Wont Pay. 

Whether the word "noon," 
which marks the beginning and 
expiration of all fire insurance 
policieB, means noon by standard 
time, or noon by solar time, is a 
question which is soon to be 
fought out in the courts of Ken- 
tucky, ia thirteen suits which 
have attracted the attention of 
tire insurance people all over the 
world. The suits are being 
brought by the Peaslee-Gaulbert 
Company and the Louisville Lead 
and Color Company of Louisville, 
and $19,940.70 of insurance 
money depends on the result. 

Now, although the policies in 
these companies all state that 
thejr were in force from noon of 
April 1, 1901, to noon of April 1, 
1902, not one of them says what 
kind of time that period o£ the 
day is to be reckoned in. In 
Louisville the solar noon is V7\ 
minutes earlier than the stand- 
ard noon, so that a fire occurring 
in the neighborhood of noon on 
the day of a policy's expiration, 
may easily be open to attack. 

The records of the Louisville 
fire department show that the fire 
that destroyed the buildings of 
the two companies was discov- 
ered at 11:45 o'clock Louisville 
standard time in the forenoon of 
April 1, last. The fire began in 
the engine room of the main fac- 
tory and spread to the two other 
buildings which were used mainly 
as warehouses. When the fire 
department recorded the time of 



the fire's discovery it figured, of 
course, by standard time. Solar 
time would, make it just two and 
a half minutes after noon. If 
noon in the poUoies means noon 
by solar time, of course the thir- 
teen companies are absolved from 
any responsibility for the loss. 
If it means noon by standard 
time, of course they must pay. 

When the insurance people got 
the claims of the companies they 
decHned to pay, and when asked 
for an explanation declared that 
noon in the policies meant noon 
by solar time. The burhed-out 
companies immediately began suit, 
and in their affidavits they say 
that not only is standard time 
the official time of the State of 
Kentucky and the city of Louis- 
ville, but it is also the time upon 
which all business engagements' 
and all domestic and social en- 
gagements are reckoned. They 
state further that they are pre- 
pared to show that in 1890 the 
city of Louisville passed an ordi- 
nance making standard time the 
official time of the city, that all 
legislation is dated according to 
standard time, and that the gov- 
ernor of the state is inaugurated 
at noon according to the same 
measurement of time. 

Solar time, state the companies, 
can be found in use in Louisville 
by only a few banking institu- 
tions which got charters many 
years ago that compel them to 
use solar time to this day. Most 
banks, they say, operate on stand- 
ard time, although they keep 
clocks going at solar time so as to 
do business on that basis if 
requested. Judging by standard 
time the plaintiffs allege their fire 
took place fifteen minutes before 
their policies expired. 

The suits will soon come to trial, 
and, of course, will be watched with 
great interest by insurance people. 



80 LONGITUDE AND TIME 

Iowa Case. An almost precisely similar case occurred 
at Creston, Iowa, September 19, 1897. In this instance 
the insurance pohcies expired " at 12 o'clock at noon," 
and the fire broke out at two and a half minutes past 
noon according to standard time, but at fifteen and one- 
half minutes before local mean solar noon. In each of 
these cases the question of whether standard time or local 
mean solar time was the accepted meaning of the term 
was submitted to a jury, and in the first instance the ver- 
dict was in favor of standard time, in the Iowa case the 
verdict was in favor of local time. 

Early Decision in England. In 1858 and thus prior to 
the formal adoption of standard time in Great Britain, it 
was held that the time appointed for the sitting of a court 
must be understood as the mean solar time of the place 
where the court is held and not Greenwich time, unless it be 
so expressed, and a new trial was granted to a defendant 
who had arrived at the local time appointed by the court 
but found the court had met by Greenwich time and the 
case had been decided against him.- 

Court Decision in Georgia. In a similar manner a court 
in the state of Georgia rendered the following opinion: 

" The only standard of time in computation of a day, or hours of a 
day, recognized by the laws of Georgia is the meridian of the sun ; and 
a legal day begins and ends at midnight, the mean time between meri- 
dian and meridian, or 12 o'clock post meridiem. An arbitrary and arti- 
ficial standard of time, fixed by persons in a certain line of business, . 
cannot be substituted at will in a certain locality ifor the standard 
recognized by the law." 

Need for Legal Time Adoption on a Scientific Basis. There 
is nothing in the foregoing decisions to determine whether 
mean local time, or the time as actually indicated by the 
sun at a particular day, is meant. Since the latter some- 



TIME USED IN VAEIOUS COUNTEIES 81 

times varies as much as fifteen minutes faster or slower 
than the average, opportunities for controversies are mul- 
tipUed when no scientifically accurate standard time is 
adopted by law. 

Even though statutes are expUcit in the definition of 
time, they are still subject to the official interpretation 
of the courts, as the following extracts show: 

Thomas Mier took out a fire insurance policy on his saloon at 
11:30 standard time, the policy being dated noon of that day. At 
the very minute that he was getting the policy the saloon caught fire 
and was burned. Ohio law makes standard time legal time, and the 
company refused to pay the $2,000 insurance on Mier's saloon. The 
case was fought through to the Supreme Court, which decided that 
"noon" meant the time the sun passed the meridian at Akron, which is 
at 11:27 standard time. The court ordered the insurance company to 
pay. — Law Notes, June, 1902. 

In the 28th Nebraska Reports a case is cited in which judgment by 
default was entered against a defendant in a magistrate's court who 
failed to make an appearance at the stipulated hour by standard time, 
but arrived within the limit by solar time. He contested the ruling 
of the court, and the supreme judiciary of the state upheld him in the 
contest, although there was a Nebraska statute making standard time 
the legal time. The court held that "at noon" must necessarily 
mean when the sun is over the meridian, and that no construction 
could reasonably interpret it as indicating 12 o'clock standard time. 

Time Used in Various Countries 

The following table is taken, by permission, largely 
from the abstracts of official reports given in Vol. IV, 
Appendix IV of the Publications of the United States 
Naval Observatory, 1905. The time given is fast or slow 
as compared with Greenwich mean solar time. 

Argentina, 4 h. 16 m. 48.2s. slow. Official time is referred to the merid- 
ian of Cordoba. At 11 o'clock, a.m., a daily signal is telegraphed 
from the Cordoba Observatory. 

JO. MATH. GBO. — 6 



82 LONGITUDE AND TIME 

Austria-Hungary, 1 h. fast. Standard time does not exist except for 
the service of railroads where it is in force, not by law, but by order 
of the proper authorities. 

Belgium. Official time is calculated from to 24 hours, zero corre- 
sponding to midnight at Greenwich. The Royal Observatory at 
Brussels communicates daily the precise hour by telegraph. 

British Empire. 

Great Britain. The meridian of Greenwich is the standard time 

meridian for England, Isle of Man, Orkneys, Shetland Islands, 

and Scotland. 
Ireland, h. 25 m. 21.1 s. slow. The meridian of Dublin is the 

standard time meridian. 
Africa (English Colonies), 2 h. fast. Standard time for Cape Colony, 

Natal, Orange River Colony, Rhodesia and Transvaal. 

Australia. 

New South Wales, Queensland, Tasmania and Victoria, 10 h. fast. 
South Australia and Northern Territory, 9 h. 30 m. fast. 

Canada. 

Alberta and Saskatchewan, 7 h. slow. 

British Columbia, 8 h. slow. 

Keewatin and Manitoba, 6 h. slow. 

Ontario and Quebec, 5 h. slow. 

New Brunswick, Nova Scotia, and Prince Edward Island, 4 h. slow. 

Chatham Island, 11 h. 30 m. fast. 

Gibraltar, Greenwich time. , 

Hongkong, 8 h. fast. 

Malta, 1 h. fast. 

New Zealand, 11 h. 30 m. fast. 

India. Local mean time of the Madras Observatory, 5 h. 20 m. 59.1 s., 
is practically used as standard time for India and Ceylon, being 
telegraphed daily all over the country; but for strictly local use it 
is generally converted into local mean time. It is proposed soon 
to adopt the standard time of 5 h. 30 m. fast of Greenwich for India 
and Ceylon, and 6 h. 30 m. fast of Greenwich for Burmah. 

Newfoundland, 3 h. 30 m. 43.6 s. slow. (Local mean time of 
St. John's.) 

Chile, 4 h. 42 m. 46.1 s. slow. The official railroad time is furnished by 
the Santiago Observatory. It is telegraphed over the country daily 
at 7 o'clock, A.M. The city of Valparaiso uses the local time, 4 h. 
46 h. 34.1 m. slow, of the observatory at the Naval School located 
there. 



TIME USED IN VARIOUS COUNTRIES 83 

China. An observatory is maintained by the Jesuit mission at 
Zikawei near Shanghai, and a time-ball suspended from a mast on 
the French Bund in Shanghai is dropped electrically precisely at 
noon each day. This furnishes the local time at the port of 
Shanghai 8 h. 5 m. 43.3 s. fast, which is adopted by the railway and 
telegraph companies represented there, as well as by the coastwise 
shipping. From Shanghai the time is telegraphed to other ports. 
The Imperial Railways of North China use the same time, taking 
it from the British gun at Tientsin and passing it on to the stations 
of the railway twice each day, at 8 o'clock a.m. and at 8 o'clock p.m. 
Standard time, 7 h. and 8 h. fast, is coming into use all along the 
east coast of China from Newchwang to Hongkong. 

Colombia. Local mean time is used at Bogota, 4 h. 56 m. 54.2 s. slow, 
taken every day at noon in the observatory. The lack of effective 
telegraphic service makes it impossible to communicate the time 
as corrected at Bogota to other parts of the country, it frequently 
taking four and five days to send messages a distance of from 50 to 
100 miles. 

Costa Rica, 5 h. 36 m. 16.9 s. slow. This is the local mean time of the 
Government Observatory at San Jos^. 

Cuba, 5 h. 29 m. 26 s. slow. The official time of the Republic is the civil 
mean time of the meridian of Havana and is used by the railroads 
and telegraph hnes of the government. The Central Meteorological 
Station gives the time daily to the port and city of Havana as well 
as to aU the telegraph offices of the Republic. 

Denmark, 1 h. fast. In Iceland, the Faroe Islands and the Danish 
West Indies, local mean time is used. 

Egypt, 2 h. fast. Standard time is sent out electrically by the standard 
clock of the observatory to the citadel at Cairo, to Alexandria, Fort 
Said and Wady-Halfa. 

Equador, 5 h. 14 m. 6.7 s. slow. The official time is that of the meridian 
of Quito, corrected daily from the National Observatory. 

France, h. 9 m. 20.9 s. fast. Legal time in France, Algeria and Tunis is 
local mean time of the Paris Observatory. Local mean time is 
considered legal in other French colonies. 

German Empire. 
Oermany, 1 h. fast. 
Kiaochau, 8 h. fast. 
Southwest Africa, 1 h. fast. 

It is proposed to adopt standard time for the following: 
Bismarck Archipelago, Carohnes, Mariane Islands and New Guinea, 
10 fast. 



84 LONGITUDE AND TIME 

German East Africa, 2 h. fast or 2 h. 30 m. fast. 
Kamerun, 1 h. fast. 

Samoa (after an understanding with the U. S.), 12 h. fast. 
Toga, Greenwich time. 

Greece, 1 h. 34 m. 52.9 s. fast. By royal decree of September 14, 1895, the 
time in common use is that of the mean time of Athens, which is 
transmitted from the observatory by telegraph to the towns of 
the kingdom. 

Holland. The local time of Amsterdam, Oh. 19 m. 32.3 s. fast is 
generally used, but Greenwich time is used by the post and telegraph 
administration and the railways and other transportation com- 
panies. The observatory at Leyden communicates the time twice 
a week to Amsterdam, The Hague, Rotterdam and other cities, 
and the telegraph bureau at Amsterdam signals the time to all 
the other telegraph bureaus every morning. 

Honduras. In Honduras the half hour nearest to the meridian of 
Tegucigalpa, longitude 87° 12' west from Greenwich, is generally 
used. Said hour, 6 h.^slow, is frequently determined at the National 
Institute by means of a solar chronometer and communicated by 
telephone to the Industrial School, where in turn it is indicated to 
the public by a steam whistle. The central telegraph office com- 
municates it to the various sub-offices of the Republic, whose 
clocks generally serve as a basis for the time of the villages, and in 
this manner an approximately uniform time is established through- 
out the Republic. 

Italy, 1 h. fast. Adopted by royal decree of August 10, 1893. This 
time is kept in all government establishments, ships of the Italian 
Navy in the ports of Italy, railroads, telegraph offices, and Italian 
coasting steamers. The hours are numbered from to 24, 
beginning with midnight. 

Japan. Imperial ordinance No. 61, of 1886: "The meridian that 
passes through the observatory at Greenwich, England, shall be 
the zero (0) meridian. Longitude shall be counted from the above 
meridian east and west up to 180 degrees, the east being positive 
and the west negative. From January 1, 1888, the time of the 
135th degree east longitude shall be the standard time of Japan." 
This is 9 h. fast. 

Imperial ordinance No. 167, of 1895: "The standard time hitherto 
used in Japan shall henceforth be called central standard time. The 
time of the 120th degree east longitude shall be the standard time 
of Formosa, the Pescadores, the Yaeyama, and the Miyako groups, 
and shall be called western standard time. This ordinance shall 
take effect from the first of January, 1896." This is 8 h. fast. 



TIME trSED IN VARIOUS COUNTRIES 85 

Korea, 8 h. 30 m. fast. Central standard time of Japan is telegraphed 
daily to the Imperial Japanese Post and Telegraph Office at Seoul. 
Before December, 1904, this was corrected by subtracting 30 m., 
which nearly represents the difference in longitude, and was then 
used by the railroads, street railways, and post and telegraph offices, 
and most of the better classes. Since December 1, 1904, the Jap- 
anese post-offices and railways in Korea have begun to use central 
standard time of Japan. In the country districts the people use 
sundials to some extent. 

Luxemburg, 1 h. fast, the legal and uniform time. 

Mexico, 6 h. 36 m. 26.7 s. slow. The National Astronomical Observatory 
of Tacubaya regulates a clock twice a day which marks the local 
mean time of the City of Mexico, and a signal is raised twice a week 
at noon upon the roof of the national palace, such signal being 
used to regulate the city's pubhc clocks. This signal, the clock at 
the central, telegraph office, and the public clock on the cathedral, 
serve a« a basis for the time used commonly by the people. The 
general telegraph office transmits this time daily to all of its branch 
offices. Not every city in the country uses this time, however, 
since a local time, very imperfectly determined, is more commonly 
observed. The following railroad companies use standard City of 
Mexico time corrected daily by telegraph: Central, Hidalgo, Xico 
and San Rafael, National and Mexican. The Central and National 
railroads correct their clocks to City of Mexico time daily by means 
of the noon signal sent out from the Naval Observatory at Wash- 
ington (see page 71) and by a similar signal from the observatory 
at St. Louis, Missouri. The Nacozari, and the Cananea, Yaqui River 
and Pacific railroads use Mountain time, 7 h. slow, and the Sonora 
railroad uses the local time of Guaymas, 7 h. 24 m. slow. 

Nicaragua, 5 h. 45 m. 10 s. slow. Managua time is issued to all public 
offices, railways, telegraph offices and churches in a zone that 
extends from San Juan del Sur, latitude 11° 15' 44" N., to EI Ocotal, 
latitude 12° 46' N., and from El Castillo, longitude 84° 22' 37" W., 
to Corinto, longitude 87° 12' 31" W. The time of the Atlantic 
ports is usually obtained from the captains of ships. 

Norway, 1 h. fast. Central European time is used everywhere through- 
out the country. Telegraphic time signals are sent out once a 
week to the telegraph stations throughout the country from the 
observatory of the Christiania University. 

Panama. Both the local mean time of Colon, 5 h. 19 m. 39 s. slow, and 
eastern standard time of the United States, 5 h. slow, are used. The 
latter time is cabled daily by the Central and South American Cable 



86 LONGITUDE AND TIME 

Company from the Naval Observatory at "Washington, and will 
probably soon be adopted as standard. 

Peru, 5 h. 9 m. 3 s. slow. There is no official time. The railroad from 
Callao to Oroya takes its time by telegraph from the noon signal at 
the naval school at Callao, which may be said to be the. standard 
time for Callao, Lima, and the whole of central Peru. The railroad 
from MoUendo to Lake Titicaca, in southern Peru, takes its time 
from ships in the Bay of MoUendo. 

Portugal, h. 36 m. 44.7 s. slow. Standard time is in use throughout 
Portugal and is based upon the meridian of Lisbon. It is estab- 
lished by the Royal Observatory in the Royal Park at Lisbon, 
and from there sent by telegraph to every railway station through- 
out Portugal having telegraphic communication. Clocks on railway 
station platforms are five minutes behind and clocks outside of 
stations are true. 

Russia, 2 h. 1 m! 18.6 s. fast. All telegraph stations use the time of the 
Royal Observatory at Pulkowa, near St. Petersburg. At railroad 
stations both local and Pulkowa time are given, from which it is 
to be inferred that for all local purposes local time is used. 

Salvador, 5 h. 56 m. 32 s. slow. The government has established a national 
observatory at San Salvador which issues time on Wednesdays and 
Saturdays, at noon, to all public offices, telegraph offices, railways, 
etc., throughout the Republic. 

Santo Domingo, 4 h. 39 m. 32 s. slow. Local mean time is used, but there 
is no central observatory and no means of correcting the time. The 
time differs from that of the naval vessels in these waters by about 
30 minutes. 

Servia, 1 h. fast. Central European time is used by the railroad, tele- 
graph companies, and people generally. Clocks are regulated by 
telegraph from Budapest every day at noon. 

Spain, Greenwich time. This is the official time for use in govern- 
mental offices in Spain and the Balearic Islands, railroad and 
telegraph offices. The hours are numbered from to 24, beginning 
with midnight. In some portions local time is still used for private 
matters. 

Sweden, 1 h. fast. Central European time is the standard in general 
use. It is sent out every week by telegraph from the Stockholm 
Observatory. 

Switzerland, 1 h. fast. Central European time is the only legal time. 
It is sent out daily by telegraph from the Cantonal Observatory at 
Neuchatel. 



TIME USED IN VARIOUS COUNTRIES 87 

Turkey. Two kinds of time are used, Turlcish and Eastern European 
time, the former for the natives and the latter for Europeans. The 
railroads generally use both, the latter for the actual running of 
trains and Turkish time-tables for the benefit of the natives. 
Standard Turkish time is used generally by the people, sunset being 
the base, and twelve hours being added for a theoretical sunrise. 
The official clocks are set daily so as to read 12 o'clock at the theo- 
retical sunrise, from tables showing the times of sunset, but the 
tower clocks are set only two or three times a week. The govern- 
ment telegraph lines use Turkish time throughout the empire, and 
St. Sophia time, 1 h. 56 m. 53 s. fast, for telegrams sent out of the 
country. 

United States. Standard time based upon the meridian of Greenwich, 
varying by whole hours from Greenwich time, is almost universally 
used, and is sent out daily by telegraph to most of the country, and 
to Havana and Panama from the Naval Observatory at Washington, 
and to the Pacific coast from the observatory at Mare Island Navy 
Yard, California. For further discussions of standard time belts in 
the United States, see pp. 66-68 and the U. S. standard time belt 
map. Insular possessions have time as follows: 
Porto Rico, 4 h. slow, Atlantic standard time. 
Alaska, 9 h. slow, Alaska standard time. 
Hawaiian Islands, 10 h. 30 m. slow, Hawaiian standard time. 
GiMm, 9 h. 30 m. fast, Guam standard time. 
Philippine Islands, 8 h. fast, Philippine standard time. 
Tutuila, Samoa, 11 h. 30 m. slow, Samoan standard time. 

Uruguay, 3 h. 44 m. 48.9 s. slow. The time in common use is the mean 
time of the meridian of the dome of the Metropolitan Church of 
Montevideo. The correct time is indicated by a striking clock in 
the tower of that church. An astronomical geodetic observatory, 
with meridian telescope and chronometers, has now been estab- 
lished and will in the future furnish the time. It is proposed to 
install a time ball for the benefit of navigators at the port of Monte- 
video. An electric time service will be extended throughout the 
country, using at first the meridian of the church and afterwards 
that of the national observatory, when constructed. 

Venezuela, 4 h. 27 m. 43.6 s. The time is computed daily at the Caracas 
Observatory from observations of the sun and is occasionally tele- 
graphed to other parts of Venezuela. The cathedral clock at Caracas 
-is corrected by means of these observations. Railway time is at 
least five minutes later than that indicated by the cathedral clock, 
which is accepted as standard by the people. Some people take 
time from the observatory flag, which always falls at noon. 



88 



LONGITUDE AND TIME 



Latitude and Longitude of Cities 

The latitude and longitude of cities in the following 
table was compiled from various sources. Where possible, 
the exact place is given, the abbreviation " " standing 
for observatory, " C " for cathedral, etc. 



Adelaide, S. Australia, Snap- 
per Point 

Aden, Arabia, Tel. Station . 

Alexandria, Egypt, Eunos Pt. 

Amsterdam, Holland, Ch. . 

Antwerp, Belgium, O. . . . 

Apia, Samoa', Ruge's Wharf 

Athens, Greece, O 

Bangkok, Siam, Old Br. Fact. 

Barcelona, Spain, Old Mole 
Light 

Batavia, Java, O 

Bergen, Norway, C 

BerUn, Germany, O 

Bombay, India, O 

Bordeaux, France, O. . . . 

Brussels, Belgium, O. . . . 

Buenos Aires, Custom House 

Cadiz, Spain, O 

Cairo, Egypt, O 

Calcutta, Ft. Wm. Semaphore 

Canton, China, Dutch Light 

Cape Horn, South Summit . 

Cape Town, S. Africa, O. 

Cayenne, Fr. Guiana, Landing 

Christiania, Norway, O. . . 

Constantinople, Turkey, C. . 

Copenhagen, Denmark, New O. 

Dublin, Ireland, O 

Edinburgh, Scotland, O. . . 

Florence, Italy, O 

Gibraltar, Spain, Dock Flag 

Glasgow, Scotland, O. . . . 

Hague, The, Holland, Ch. . 

Hamburg, Germany, O. . . 

Havana, Cuba, Morro Lt. H. 

Hongkong, China, C. . . . 





Latitude 


Longitude from 
Greenwich 


34° 


46' 


50" S 


138° 


30' 


39" E 


12° 


46' 


40" N 


44° 


58' 


58" E 


31° 


11' 


43" N 


29° 


51' 


40" E 


52° 


22' 


30" N 


4° 


53' 


04" E 


■ 51° 


12' 


28" N 


4° 


24' 


44" E 


13° 


48' 


56" S 


171° 


44' 


56" W 


37° 


58' 


21" N 


23° 


43' 


55" E 


13° 


44' 


20" N 


100° 


28' 


42" E 


41° 


22' 


10" N 


2° 


10' 


55" E 


6° 


07' 


40" N 


106° 


48' 


25" E 


60° 


23' 


37" N 


5° 


20' 


15" E 


52° 


30' 


17" N 


13° 


23' 


44" E 


18° 


53' 


45" N 


72° 


48' 


58" E 


44° 


50' 


07" N 


00° 


31' 


23" W 


50° 


51' 


11" N 


4° 


22' 


18" E 


34° 


36' 


30" S 


58° 


22' 


14" W 


36° 


27' 


40" N 


6° 


12' 


20" W 


30° 


04' 


38" N 


31° 


17' 


14" E 


22° 


33' 


25" N 


88° 


20' 


11" E 


23° 


06' 


35" N 


113° 


16' 


34" E 


55° 


58' 


41" S 


67° 


16' 


15" W 


33° 


56' 


03" S 


18° 


28' 


40" E 


4° 


56' 


20" N 


52° 


20' 


25" W 


59° 


54' 


44" N 


10° 


43' 


35" E 


41° 


00' 


16" N 


28° 


58' 


59" E 


55° 


41' 


14" N 


12° 


34' 


47" E 


53° 


23' 


13" N 


6° 


20' 


30" W 


55° 


57' 


23" N 


3° 


10' 


64" W 


43° 


46' 


04" N 


11° 


15' 


22" E 


36° 


07' 


10" N 


5° 


21' 


17" W 


55° 


52' 


43" N 


4° 


17' 


39" W 


62° 


04' 


40" N 


4° 


18' 


30" E 


53° 


33' 


07" N 


9° 


58' 


25" E 


23° 


09' 


21" N 


82° 


21' 


30" W 


21° 


16' 


52" N 


114° 


09' 


31" E 



LATITUDE AND LONGITUDE OF CITIES 



89 





Latitude 


Longitude from 
Greenwich 


Jerusalem, Palestine, Ch. 


31° 


46' 


45" N 


■ 35° 


13' 


25" E 


Leipzig, Germany, 0. . . 


51° 


20' 


06" N 


12° 


23' 


30" E 


Lisbon, Portugal, 0. (Royal 


) 38° 


42' 


31" N 


9° 


11' 


10" W 


Liverpool, England, 0. 


53° 


24" 


04" N 


3° 


04' 


16" W 


Madras, India, 


13° 


04' 


06" N 


80° 


14' 


51" B 


Marseilles, France, New 0. 


43° 


18' 


22" N 


5° 


23' 


43" E 


Melbourne, Victoria, O. . 


37° 


49' 


53" S 


144° 


58' 


32" E 


Mexico, Mexico, 0. . . . 


19° 


26' 


01" N 


99° 


06' 


39" W 


Montevideo, Uruguay, C. 


34° 


54' 


33" S 


56° 


12' 


15" W 


Moscow, Russia, 0. . . . 


55° 


45' 


20" N 


37° 


32' 


36" E 


Munich, Germany, 0. . . 


48° 


08' 


45" N 


11° 


36' 


32" E 


Naples, Italy, 


40° 


51' 


46" N 


14° 


14' 


44" E 


Panama, Cent. Am., C. . . 


8° 


57' 


06" N 


79° 


32' 


12" W 


Para, Brazil, Custom H. . 


1° 


26' 


59" 8 


48° 


30' 


01" W 


Paris, France, 


48° 


50' 


11" N 


2° 


20' 


14" E 


Peking, China 


39° 


56' 


00" N 


116° 


28' 


54" E 


Pulkowa, Russia, O . . . 


59° 


46' 


19" N 


30° 


19' 


40" E 


Rio de Janeiro, Brazil, 0. 


22° 


54' 


24" S 


43° 


10' 


21" W 


Rome, Italy, O 


41° 


53' 


54" N 


12° 


28' 


40" E 


Rotterdam, HoU., Time Bal 


51° 


54' 


30" N 


4° 


28' 


50" E 


St. Petersburg, Russia, set 


3 












Pulkowa 














Stockholm, Sweden, 0. . . 


59° 


20' 


35" N 


18° 


03' 


30" E 


Sydney, N. S. Wales, 0. . 


33° 


51' 


41" S 


151° 


12' 


23" E 


Tokyo, Japan, 


35° 


39' 


17" N 


139° 


44' 


30" E 


Valparaiso, Chile, Light Hou 


se 33° 


01' 


30" S 


71° 


39' 


22" W 



United States 



Aberdeen, S. D., N. N. & I. S. 
Albany, N. Y., New O. . . 
Ann Arbor, Mich., O. . . . 

Annapolis, Md., O 

Atlanta, Ga., Capitol . . . 
Attu Island, Alaska, Chi- 

chagoff Harbor 

Augusta, Me., Baptist Ch. . 
Austin, Tex. ........ 

Baltimore, Md., Wash. Mt. . 
Bangor, Me., Thomas Hill . 
Beloit, Wis., College .... 

Berkeley, Cal., O 

Bismarck, N. D 

Boise, Idaho, Ast. Pier . . 
Boston, Mass., State House 



45° 27' 

42° 39' 

42° 16' 

38° 58' 

33° 45' 



52° 
44° 
32° 
39° 



56' 
18' 
00' 
17' 



44° 48' 

42° 30' 

37° 52' 

46° 49' 

43° 35' 

42° 21' 



50" N 
13" N 
48" N 
53" N 
19" N 

01" N 
52" N 
40" N 
48" N 
23" N 
13" N 
24" N 
12" N 
58" N 
28" N 



98° 
73° 
83° 
76° 
84° 

173° 

69° 

100° 

76° 

68° 

89° 

122° 

100° 

116° 

71° 



28' 
46' 
43' 
29' 
23' 

12' 
46' 
27' 
36' 
46' 
1' 
15' 
45' 
13' 
03' 



45" W 
42" W 
48" W 
08" W 
29" W 

24" E 
37" W 
35" W 
59" W 
59" W 
46" W 
41" W 
08" W 
04" W 
50" W 



90 



LONGITUDE AND TIME 





Latitude 


Longitude from 
Greenwich 


42= 


53' 


03" N 


78° 


52' 


42" W 


32° 


4T 


44" N 


79° 


52' 


58" W 


41° 


07' 


47" N 


104° 


48' 


52' W 


41° 


50' 


01" N 


87° 


36' 


36" W 


39° 


08' 


19" N 


84° 


26' 


00" W 


41° 


30' 


02" N 


81° 


42' 


10" W 


33° 


59' 


12" N 


81° 


00' 


12" W 


39° 


57' 


40" N 


82° 


59' 


40" W 


43° 


11' 


48" N 


71° 


32' 


30" W 


44° 


22' 


34" N 


103° 


43' 


19" W 


39° 


40' 


36" N 


104° 


59' 


23" W 


41° 


35' 


08" N 


93° 


37' 


30" W 


42° 


20' 


00" N 


83° 


02' 


54" W 


46° 


48' 


00" N 


92° 


06' 


10" W 


42° 


07' 


53" N 


80° 


05' 


51" W 


46° 


52' 


04" N 


96° 


47' 


11" W 


29° 


18' 


17" N 


94° 


47' 


26" W 


35° 


51' 


48" N 


100° 


26' 


24" W 


41° 


45' 


59" N 


72° 


40' 


45" W 


46" 


35' 


36" N 


111° 


52' 


45" W 


21° 


18' 


12" N 


157° 


51' 


34" W 


39° 


47' 


00" N 


86° 


05' 


00" W 


31° 


16' 


00" N 


91° 


36' 


18" W 


30° 


19' 


43" N 


81° 


39' 


14" W 


39° 


06' 


08" N 


94° 


35' 


19" W 


24° 


32' 


58" N 


81° 


48' 


04" W 


42° 


43' 


56" N 


84° 


33' 


23" W 


38° 


02' 


25" N 


84° 


30' 


21" W 


40° 


55' 


00" N 


96° 


52' 


00" W 


34° 


40' 


00" N 


92° 


12' 


00" W 


34° 


03' 


05" N 


118° 


14' 


32" W 


38" 


15' 


08" N 


85° 


45' 


29" W 


42" 


22' 


00" N 


71° 


04' 


00" W 


43" 


04' 


37" N 


89° 


24' 


27" W 


14" 


35' 


31" N 


120° 


58' 


03" E 


35° 


08' 


38" N 


90° 


03' 


00" W 


43° 


02' 


32" N 


87° 


54' 


18" W 


44" 


58' 


38" N 


93° 


14' 


02" W 


43" 


49' 


00" N 


98° 


00' 


14" W 


30" 


41' 


26" N 


88° 


02' 


28" W 


32" 


22' 


46" N 


86° 


17' 


57" W 


36" 


08' 


54" N 


86° 


48' 


GO" W 


40" 


44' 


06" N 


74° 


10' 


12" W 


41" 


18' 


28" N 


72° 


55' 


45" W 


29" 


57' 


46" N 


90° 


03' 


28" W 


40" 


42' 


44" N 


74° 


00' 


24"-W 


44" 


27' 


42" N 


93° 


08' 


57" W 



Buffalo, N. Y 

Charleston, S. C, Lt. House 
Cheyenne, Wyo., Ast. Sta. . 

Chicago, 111., O 

Cincinnati, Ohio 

Cleveland, Ohio, Lt. H. . . 

Columbia, S. C 

Columbus, Ohio 

Concord, N. H 

Deadwood, S. D., P. O. . . 

Denver, Col., O 

Des Moines, Iowa 

Detroit, Mich 

Duluth, Minn 

Erie, Pa., Waterworks . . . 
Fargo, N. D., Agri. College . 

Galveston, Tex., C 

Guthrie, Okla 

Hartford, Conn 

Helena, Mont. 

Honolulu, Sandwich Islands 

Indianapolis, Ind 

Jackson, Miss 

Jacksonville, Fla., M. E. Ch. 

Kansas City, Mo 

Key West, Fla., Light House 
Lansing, Mich., Capitol . . 
Lexington, Ky., Univ. . . . 

Lincoln, Neb 

Little Rock, Ark 

Los Angeles, Cal., Ct. House 

Louisville, Ky 

Lowell, Mass 

Madison, Wis., O 

Manila, Luzon, C 

MemJDhis, Tenn. 

Milwaukee, Wis., Ct. House . 
Minneapolis, Minn., O. . . . 

Mitchell, S. D; 

Mobile, Ala., Epis. Church . 

Montgomery, Ala 

Nashville, Tenn., O 

Newark, N. J^^ M. E. Ch. . . 
New Haven, Conn., Yale . . 
New Orleans, La., Mint. . . 
New York, N. Y., City Hall. 
Northfield, Minn., O. . . . 



LATITUDE AND LONGITUDE OF CITIES 



91 



Ogden, Utah, O 

Olympia, Wash 

Omaha, Neb 

Pago Pago, Samoa .... 

Philadelphia, Pa. State House 

Pierre, S. D., Capitol. . . . 

Pittsburg, Pa 

Point Barrow (highest lati- 
tude in the United States) 

Portland, Ore 

Princeton, N. J., O 

Providence, R. I., Unit. Cli. 

Raleigh, N. C 

Richmond, Va., Capitol . . 

Rochester, N. Y., O 

Sacramento, Cal 

St. Louis, Mo. ....... 

St. Paul, Minn 

San Francisco, Cal., C. S. Sta. 
San Juan, Porto Rico, Morro 

Light House 

Santa Fe, N. M 

Savannah, Ga., Exchange 
Seattle, Wash., C. S. Ast. Sta. 
Sitka, Alaska, Parade Ground 

Tallahassee, Fla 

Trenton, N. J. Capitol . . . 

Virginia City, Nev 

Washington, D. C, O. . . . 

Wheeling, W. Va 

Wilmington, Del., Town Hall 
Winona, Minn 



Latitude 


Longitude from 
Greenwicli 


41° 


13' 


08" N 


111° 


59' 


45" W 


47° 


03' 


00" N 


122° 


57' 


00" W 


41° 


16' 


50" N 


95° 


57' 


33" W 


14° 


18' 


06" S 


170° 


42' 


31" W 


39° 


56' 


53" N 


75° 


09' 


03" W 


44° 


22' 


50" N 


100° 


20' 


26" W 


40° 


26' 


34" N 


80° 


02' 


38" W 


71° 


27' 


00" N 


156° 


l.-^' 


00" W 


45° 


30' 


00" N 


122° 


40' 


30" W 


40° 


20' 


58" N 


74° 


39' 


24" W 


41° 


49' 


28" N 


71° 


24' 


20" W 


35° 


47' 


00" N 


78° 


40' 


00" W 


37° 


32' 


19" N 


77° 


27' 


02" W 


43° 


09' 


17" N 


77° 


35' 


27" W 


38° 


33' 


38" N 


121° 


26' 


00" W 


38° 


38' 


04" N- 


90° 


12' 


16" W 


44° 


52' 


56" N 


93° 


05' 


00" W 


37° 


47' 


55" N 


122° 


24' 


32" W 


18° 


28' 


56" N 


66° 


07' 


28" W 


35° 


41' 


19" N 


105° 


56' 


45" W 


32° 


04' 


52" N 


81° 


05' 


26" W 


47° 


35' 


54" N 


122° 


19' 


59" W 


57° 


02' 


52" N 


135° 


19' 


31" W 


30° 


25' 


00" N 


84° 


18' 


00" W 


40° 


13' 


14" N 


74° 


46' 


13" W 


39° 


17' 


36" N 


119° 


39' 


06" W 


38° 


53' 


39" N 


77° 


03' 


06" W 


40° 


05' 


16" N 


80° 


44' 


30" W 


39° 


44' 


27" N 


75° 


33' 


03" W 


44° 


04' 


00" N 


91° 


30' 


00" W 



CHAPTER V 

CIRCUMNAVIOATION AND TIME 

Magellan's Fleet. When the sole surviving ship of 
Magellan's fleet returned to Spain in 1522 after having 
circumnavigated the globe, it is said that the crew were 
greatly astonished that their calendar and that of the 
Spaniards did not correspond. They landed, according to 
their own reckoning, on September 6, but were told it was 
September 7. At first they thought they had made a mis- 
take, and some time elapsed before they reahzed that they 
had lost a day by going around the world with the sun. 
Had they traveled toward the east, they would have 
gained a day, and would have recorded the same date as 
September 8. 

"My pilot is dead of scurvy: may 
I ask the longitude, time and day? " 
The first two given and compared; 
The third, — the commandante stared! 

"The first of June? I make it second," 
Said the stranger, ''Then you've wrongly reckoned 1" 
— Bret Harte, in The Lost Galleon. 

The explanation of this phenomenon is simple. In 
traveling westward, in the same way with the sun, one's 
days are lengthened as compared with the day at any 
fixed place. When one has traveled 15° westward, at what- 
ever rate of speed, he finds his watch is one hour behind 
the time at his starting point, if he changes it according 
to the sun. He has thus lost an hour as compared with 

92 



WESTWARD TRAVEL— DAYS ARE LENGTHENED 93 

the time at his starting point. T^ter he has traveled 15° 
farther, he will set his watch back two hours and thus 
record a loss of two hours. And so it continues through- 
out the twenty-four belts of 15° each, losing one hour in 
each belt; by the time he arrives at his starting point 




Fig. 30 

again, he has set his hour hand back twenty-four hours 
and has lost a day. 

Westward Travel - — Days are Lengthened. To make this 
clearer, let us suppose a traveler starts from London Mon- 
day noon, January 1st, travehng westward 15° each day. 
On Tuesday, when he finds he is 15° west of London, he 
sets his watch back an hour. It is then noon by the sun 
where he is. He says, " I left Monday noon, it is now 



94 CIRCUMNAVIGATION AND TIME 

Tuesday noon; therefore I have been out one day." The 
tower clock at London and his chronometer set with it, 
however, indicate a different view. They say it is Tuesday, 
1 o'clock, P.M., and he has been out a day and an hour. 
The next day the process is repeated. The traveler, hav- 
ing covered another space of 15° westward, sets his watch 
back a second hour and says, " It is Wednesday noon and 
I have been out just two days." The London clock, how- 
ever, says Wednesday, 2 o'clock, p.m. — two days and 
two hours since he left. The third day this occurs again, 
the traveler losing a third hour ; and what to him seems 
three days, Monday noon to Thursday noon, is in reality 
by London time three days and three hours. Each of his 
days is really a Httle more than twenty-four hours long, 
for he is going with the sun. By the time he arrives at 
London again he finds what to him was twenty-four days 
is, in reality, twenty-five days, for he has set his watch 
back an hour each day for twenty-four days, or an entire 
day. To have his calendar correct, he must omit a day, 
that is, move the date ahead one day to make up the date 
lost from his reckoning. It is obvious that this will be 
true whatever the rate of travel, and the day can be omitted 
from his calendar anywhere in the journey and the error 
corrected. 

Eastward Travel — Days are Shortened. Had our trav- 
eler gone eastward, when he had covered 15° of longitude 
he would set his watch ahead one hour and then say, 
" It is now Tuesday noon. I have been out one day." 
The London clock would indicate 11 o'clock, a.m., of 
Tuesday, and thus say his day had but twenty-three 
hours in it, the traveler having moved the hour hand 
ahead one space. He has gained one hour. The second 
day he would g&in another hour, and by the time he arrived 



THE INTERNATIONAL BATE LINE 



95 



at London again, he would 
have set his hour hand ahead 
twenty-four hours or one 
full day. To correct his cal- 
endar, somewhere on his voy- 
age he would have to repeat 
a day. 

The International Date 
Line. It is obvious from 
the foregoing explanation 
that somewhere and some- 
time in circumnavigation, a 
day must be omitted in trav- 
eling westward and a day 
repeated in traveling east- 
ward. Where and when the 
change is made is a mere 
matter of convenience. The 
theoretical location of the 
date line commonly used is 
the 180th meridian. TJiis 
line where a traveler's cal- 
endar needs changing varies 
as do the boundaries of the 
standard time belts and for 
the same reason. While 
the change could be made 
at any particular point on 
a parallel, it would make a 
serious inconvenience were 
the change made in some 
places. Imagine, for ex- 
ample, the 90th meridian. 





KERMADEC IS. 



ZEALAND/^' 



IColE 



t CHATHAM 



165' 



W 



Fig- 31 



96 CIRCUMNAVIGATION AND TIME 

west of Greenwich, to be the line used. When it was 
Sunday in Chicago, New York, and other eastern points, 
it would be Monday in St. Paul, Kansas City, and western 
points. A traveler leaving Minneapohs on Sunday night 
would arrive in Chicago on Sunday morning and thus 
have two Sundays on successive days. Our national holi- 
days and elections would then occur on different days 
in different parts of the country. To reduce to the 
minimiun such inconveniences as necessarily attend chang- 
ing one's calendar, the change is made where there is a 
relatively small amount of travel, away out in the Pacific 
Ocean. Going westward across this lino one must set his 
calendar ahead a day; going eastward, back a day. 

As shown in Figures 31 and 32, this Une begins on the 
180th meridian far to the north, sweeps to the eastward 
around Cape Deshnef, Russia, then westward beyond the 
180th meridian seven degrees that the Aleutian islands 
may be to the east of it and have the same day as 
continental United States; then the line extends to the 
180th meridian which it follows southward, sweeping 
somewhat eastward to give the Fiji and Chatham islands 
the same day as Australia and New Zealand. The follow- 
ing is a letter, by C. B. T. Moore, commander, U. S. N., 
Governor of Tutuila, relative to the accuracy of the map 
in this book : 

Pago-Pago, Samoa, December 1, 1906. 

Dear Sir: — The map of your Mathematical Geography is 
correct in placing Samoa to the east of the international date line. 
The older geographies were also right in placing these islands west 
of the international date line, because they used to keep the same date 
as Australia and New Zealand, which are west of the international 
date line. 

The reason for this mistake is that when ther London Missionary 
Society sent its missionaries to Samoa they were not acquainted with 



THE INTERNATIONAL DATE LINE 97 

the trick of changing the date at the 180th meridian, and so carried 
into Samoa, which was east of the date Hne, the date they brought 
with them, which was, of course, one day ahead. 

This false date was in force at the time of my first visit to Samoa, 
in 1889. While I have no record to show when the date was 
corrected, I beUeve that it was corrected at the time of the annexa- 
tion of the Samioan Islands by the United States and by Germany. 
The date in Samoa is, therefore, the same date as in the United 
States, and is one day behind what it is in Australia and New Zealand ; 
Example: To-day is the 2d day of December in Auckland, and the 
1st day of December in Tutuila. 

Very respectfully, C. B. T. Moore, 

Commander, U. S. Navy, 
Governor. 

Mb. Willis E. Johnson, 

Vice President Northern Normal and Industrial School, 
Aberdeen, South Dakota. 

"It is forttmate that the 180th meridian falls where it 
does. From Siberia to the Antarctic continent this 
imaginary line traverses nothing but water. The only 
land which it passes at all near is one of the archipelagoes 
of the south Pacific; and there it divides but a handful of 
volcanoes and coral reefs from the main group. These 
islands are even more unimportant to the world than 
insignificant in size. Those who tenant them are few, 
and those who are bound to these few still fewer. . . . 
There, though time flows ceaselessly on, occurs that 
unnatural yet imavoidable jump of twenty-four hours; 
and no one is there to be startled by the fact, — no one to 
be perplexed in trying to reconcile the two incongruities, 
continuous time and discontinuous day. There- is nothing 
but the ocean, and that is tenantless. . . . Most fortunate 
'was it, indeed, that opposite the spot where man was most 
destined to think there should have been placed so little 
to think about." * 

* From CJioson, by Percival Lowell. 

JO. MATH. GEO. — 7 



98 CIRCUMNAVIGATION AND TIME 

Where Days Begin. When it is 11:30 o'clock, p.m., on 
Saturday at Denver, it is 1 : 30 o'clock, a.m., Sunday, at 
New York. It is thus evident that parts of two days 
exist at the same time on the earth. Were one to travel 
around the earth with the sun and as rapidly it would be 
perpetually noon. ^Vhen he has gone around once, one 
day has passed. Where did that day begin? Or, sup- 
pose we wished to be the first on earth to hail the new 
year, where could we go to do so? The midnight line, 
just opposite the sun, is constantly bringing a new day 
somewhere. Midnight ushers in the new year at Chicago. 
Previous to this it was begun at New York. Still east 
of this. New Year's Day began some time before. If we 
keep going around eastward we must surely come to 
some place where New Year's Day was first counted, or we 
shall get entirely around to New York and find that the 
New Year's Day began the day before, and this midiiight 
wotild commence it again. As previously stated, the date 
hne commonly accepted nearly coincides with the 180th 
meridian. Here it is that New Year's Day first dawns 
and each new day begins. 

The Total Duration of a Day. While a day at any 
particular place is twenty-four hours long, each day lasts 
on earth at least forty-eight hours. Any given day, say 
Christmas, is first counted as that day just west of the 
date hne. An hour later Christmas begins 15° west of 
that hne, two hours later it begins 30° west of it, and so on 
around the globe. The people just west of the date line 
who first hailed Christmas have enjoyed twelve hours of 
it when it begins in England, eighteen hours of it when 
it begins in central United States, and twenty-four hours 
of it, or the whole day, when it begins in western Alaska, 
just east of the date fine. Christmas,' then, has existed 



COIWUSION OF TRAVELERS 99 

twenty-foTir hours on the globe, but having just begun in 
western Alaska, it will tarry twenty-four hours longer 
among mankind, making forty-eight hours that the day 
blesses the earth. 

If the date line followed the meridian 180° without 
any variation, the total duration of a day would be exactly 
forty-eight hours as just explained. But that Une is quite 
irregular, as previously described and as shown on the 
map. Because of this irregularity of the date hne the 
same day lasts somewhere on earth oyer forty-nine hours. 
Suppose we start at Cape Deshnef, Siberia, longitude 169° 
West, a moment after midnight of the 3d of July. The 
4th of July has begim, and, as midnight sweeps around 
westward, successive places see the beginning of this day. 
When it is the 4th in London it has been the 4th at 
Cape Deshnef twelve hours and forty-four minutes. When 
the glorious day arrives at New York, it has been seven- 
teen hoiirs and forty-four minutes since it began at Cape 
Deshnef. When it reaches our most western point on this 
continent, Attn Island, 173° E., it has been twenty-five 
hours and twelve minutes since it began at Cape Deshnef. 
Since it wUl last twenty-four hours at Attu Island, forty- 
nine hours and twelve minutes will have elapsed since the 
beginning of the day vmtil the moment when all places 
on earth cease to count it that day. 

When Three Days Coexist. Portions of three days 
exist at the same time between 11:30 o'clock, a.m., and 
12:30 o'clock, p.m., London time. When it is Monday 
noon at London, Tuesday has begun at Cape Deshnef, 
but Monday morning has not yet dawned at Attu Island; 
nearly half an hour of Sunday still remains there. 

Confusion of Travelers. Many stories are told of the 
confusion to travelers who pass from places reckoning 



100 



CIRCUMNAVIGATION AND TIME 



one day across this line, to places having a different day. 
" If it is such a deadly sin to work on Sunday, one or. the 
other of Mr. A and Mr. B coining, one from the east, 
the other from the west of the 180th meridian, must, if 
he continues his daily vocations, be in a bad way. Some 
of our people in the Fiji are in this unenviable position, 
as the hne 180° passes through Loma-Loma. I went 
from Fiji to Tonga in Her Majesty's ship Nymph and 
arrived at our destination on Sunday, according to our 
reckoning from Fiji, but on Saturday, according to the 
proper computation west from Greenwich. We, how- 




Fig. 3a 



ever, found the natives all keeping Sunday. On my ask- 
ing the missionaries about it they told me that the mis- 
sionaries to that group and Samoa having come from 



CONFUSION OF TRAVELERS 101 

the westward, had determined to observe their Sabbath 
day, as usual, so as not to subject the natives to any 
puzzle, and agreed to put the dividing Une farther off,, 
between them and Hawaii, somewhere in the broad ocean 
where no metaphysical natives or ' intelligent ' Zulus 
could cross-question them." * 

" A party of missionaries bo\ind from China, saiUng 
west, and nearing the hne without their knowledge, on 
Saturday posted a notice in the cabin announcing that 
' To-morrow being Sunday there wiU be ser\dces in this 
cabin at 10 a.m.' The following morning at 9, the captain 
tacked up a notice declaring that ' This being Monday 
there will be no services in this cabin tliis morning.' " 

It should be remembered that this Une, called " inter- 
national," has not been adopted by all nations as a hard 
and fast line, making it absolutely necessary to change 
the date the moment it is crossed. A ship sailing, say, 
from Honolulu, which has the same day as North America 
and Europe, to Manila or Hongkong, having a day later, 
may make the change in date at any time between these 
distant points; and since several days elapse in the pas- 
sage, the change is usually made so as to have neither 
two Sundays in one week nor a week without a Sunday. 
Jiist as the traveler in the United States going from a 
place having one time standard to a place having a dif- 
ferent one would find it necessary to change his watch 
but could make the change at any time, so one passing 
from a place having one day to one reckoning another, 
could suit his convenience as to the precise spot where 
he make the change. This statement needs only the 

* Mr. E: L. LayarcJ, at the British Consulate, Noumea, New Cale- 
donia, as quoted in a pamphlet on the International Date Line by 
Henry CoUins. 



102 CIRCUMNAVIGATION AND TIMfi 

modification that as all events on a ship must be regu- 
lated by a common timepiece, changed according to 
longitude, so the community on board in order to adjust 
to a common calendar must accept the change when made 
by the captain. ' 

Origin and Change of Date Line. The origin of this 
line is of considerable interest. The day adopted in any 
region depended upon the direction from which the 
people came who settled the country. For example, 
people who went to Australia, Hongkong, and other Eng- 
hsh possessions in the Orient traveled around Africa or 
across the Mediterranean. They thus set their watches 
ahead an hour for every 15°. " For two centuries after 
the Spanish settlement the trade of Manila with the 
western world was carried on via Acapulco and Mexico " 
(Ency. Brit.). Thus the time which obtained in the 
Phihppines was found by setting watches backwards an 
hour for every 15° and so it came about that the calendar 
of the Philippines was a day earlier than that of Australia, 
Hongkong, etc. The date line at that time was very indefi- 
nite and irregular. In 1845 by a decree of the Bishop of 
Manila, who was also Governor-General, Tuesday, Decem- 
ber 31, was stricken from the calendar; the day after 
Monday, December 30, was Wednesday, January 1, 1846. 
This cutting the year to 364 days and the week to 6 days 
gave the Philippines the same day as other Asiatic places, 
and shifted the date line to the east of that archipelago. 
Had this change never been made, all of the possessions 
of the United States would have the same day. 

For some time after the acquisition of Alaska the people 
living there, formerly citizens of Russia, used the day later 
than ours, and also used the Russian or JuUan calendar, 
twelve days later than ours. As people moved there from 



PROBLEM 103 

the United States, our system gradually was extended, but 
for a time both systems were in vogue. This made affairs 
confusing, some keeping Sunday when others reckoned the 
same day as Saturday and counted it as twelve days later 
in the calendar, New Year's Day, Christmas, etc., coming 
at different tinies. Soon, however, the American system 
prevailed to the entire exclusion of the Russian, the 
inhabitants repeating a day, and thus having eight days, 
in one week. \A'Tiile the Russians in their churches in 
Alaska are celebrating the Holy Mass on our Sunday, 
their brethren in Siberia, not far away, and in other parts 
of Russia, are busy with Monday's duties. 

Problem. The following problem, with local varia- 
tions, went the rounds in the United States in 1898. 
'■Assuming it was 5 a.m., Sunday, May 1, when the 
naval battle of Manila was begun, what time was it in 
Milwaukee, Wis. (87° 54' W.)? " The following answer was 
asserted to be correct. " About seven minutes after the 
town clock in Milwaukee struck three, Saturday p.m., 
April 30, the battle of Manila began." Show that the 
foregoing answer is incorrect, the town clock using stand- 
ard time, Dewey using local time of about 120° east. 



CHAPTER VI 

the earth's revolution 
Proofs of Revolution 

For at least 2400 years the theory of the revolution 
of the earth around the sun has been advocated, but only 
in modern times has the fact been demonstrated beyond 
successful contradiction. The proofs rest upon three sets 
of astronomical observations, all of which are of a delicate 
and abstruse character, although the underljdng principles 
are easily understood. 

Aberration of Light. When rain is falling on a calm day 
the drops will strike the top of one's head if he is stand- 
ing still in the rain; but if one moves, the direction of the 
drops will seem to have changed, striking one in the face 
more and more as the speed is increased (Fig. 33). Now 
light rays from the sun, a star, or other heavenly body, 
strike the earth somewhat slantingly, because the earth is 
moving around the sun at the rate of over a thousand 
miles per minute. Because of thir: fact the astronomer 
must tip his telescope slightly to the east of a star in order 
to see it when the earth is in one side of its orbit, and to 
the west of it when in the opposite side of the orbit. The 
necessity of this tipping of the telescope will be apparent 
if we imagine the rays passing through the telescope are 
Uke raindrops falling through a tube. If the tube is car- 
ried forward swiftly enough the drops will strike the sides 
of the tube, and in order that they may pass directly 
through it, the tube must be tilted forward somewhat, 

104 



ABERRATION OF LIGHT 105 

the amount varying with (a) the rate of its onward motion, 
and (b) the rate at which the raindrops are falling. 

Since the telescope must at one time be tilted one way 
to see a star and at another season tilted an equal amount 
in the opposite direction, each star thus seems to move 




Fig. 33 

about in a tiny orbit, varying from a circle to a straight 
line, depending upon the position of the star, but in every 
case the major axis is 41", or twice the greatest angle at 
which the telescope must be tilted forward. 

Each of the milHons of stars has its own apparent aber- 
rational orbit, no two being exactly alike in form, imless 
the two chance to be exactly the same distance from the 
plane of the earth's orbit. Assuming that the earth 



106 THE EARTH'S REVOLUTION 

revolves around the sun, the precise form of this aberra- 
tional orbit of any star can be calculated, and observation 
invariably confirms the calculation. Rational minds can- 
not conceive that the milhons of stars, at varying dis- 
tances, can all actually have these peculiar annual motions, 
six months toward the earth and six months from it, in 
addition to the other motions which many of them (and 
probably all of them, see pp. 265-267) have. The dis- 
covery and explanation of these facts in 1727 by James 
Bradley (see p. 278), the English Astronomer Royal, forever 
put at rest all disputes as to the revolution of the earth. 

Motion in the Line of Sight. If you have stood near by 
when a swiftly moving train passed with its bell ringing, - 
you may have noticed a sudden change in the tone of the 
bell; it rings a lower note immediately upon passing. The 
pitch of a note depends upon the rate at which the" sound 
waves strike the ear; the more rapid they are, the higher 
is the pitch. Imagine a boy throviing chips * into a river 
at a uniform rate while walking down stream toward a 
bridge and then while walking upstream away from the 
bridge. The chips will be closer together as they pass 
under the bridge when the boy is walking toward it than 
when he is walking away from it. In a similar way the 
sound waves from the bell of the rapidly approaching 
locomotive accumulate upon the ear of the listener, and 
the pitch is higher than it would be if the train were sta- 
tionary, and after the train passes the soimd waves will be 
farther apart, as observed by the same person, who will 
hear a lower note in consequence. 

Color varies with Rate of Vibration. Now in a precisely 
similar manner the colors in a ray of hght vary in the rate 

* This illustration is adapted from Todd's New Astronomy, p. 
432. 



MOTION IN THE LiNE OF SlGHT 10? 

of vibration. The violet is the most rapid,* indigo about 
one tenth part slower, blue slightly slower still, then green, 
yellow, orange, and red. The spectroscope is an astro- 
nomical instrument which spreads out the line of light 
from a celestial body into a band and breaks it up into 
its several colors. If a ringing bell rapidly approaches us, 
or if we approach it, the tone of the bell sounds higher 
than if it recedes from us or if we recede from it. If we 
rapidly approach a star, or a star approaches us, its color 
shifts toward the violet end of the spectroscope; and if we 
rapidly recede from it, or it recedes from us, its color shifts 
toward the red end. Now year after year the thousands 
of stars in the vicinity of the plane of the earth's pathway 
show in the spectroscope this change toward violet at one 
season and toward red at the opposite season. The far- 
ther from the plane of the earth's orbit a star is located, 
the less is this annual change in color, since the earth 
neither approaches nor recedes from stars toward the 
poles. Either the stars near the plane of the earth's 
orbit move rapidly. toward the earth at one season, gradu- 
ally stop, and six months later as rapidly recede, and stars 
away from this plane approach and recede at rates dimin- 
ishing exactly in proportion to their distance from this 
plane, or the earth itself swiftly moves about the sun. 
Proof of the Rotation of the Earth. The same set of 

* The rate of vibration per second for each of the colors in a ray 
of light is as follows: 

Violet .... 756.0 x 10'' Yellow 508.8 x 10" 

Indigo .... 698.8 X 10" Orange .... 457.1 x 10" 

Blue 617.1 X ro»2 Red 393.6 x 10" 

Green .... 569.2 x 10" 

Thus the violet color has 756 . millions of millions of vibrations each 
second; indigo, 698.8 miUions of millions, etc. 



108 THE EARTH'S EEVOLrxlON 

facts and reasoning applies to the rotation of the earth. 
In the evening a star in the east shows a color approaching 
the violet side of the spectroscope, and this gradually shifts 
toward the red during the night as the star is seen higher 
in the sky, then nearly overhead, then in the west. Now 
either the star swiftly approaches the earth early in the 
evening, then gradually pauses, and- at midnight begins to 
go away from the earth faster and faster as it approaches 
the western horizon, or the earth rotates on its axis, 
toward a star seen in the east, neither toward nor from it 
when nearly overhead, and away from it when seen near 
the west. Since the same star rises at different hours 
throughout the year it would have to fly back and forth 
toward and from the earth, two trips every day, varying 
its periods according to the time of its rising ^nd setting. 
Besides this, when a star is rising at Calcutta it shows the 
violet tendency to observers there (Calcutta is rotating 
toward the star when the star is rising), and at the same 
moment the same star is setting at New Orleans and thus 
shows a shift toward the red to observers there. Now the 
distant star cannot possibly be actually rapidly approach- 
ing Calcutta and at the same time be as rapidly receding 
from New Orleans. The spectroscope, that wonderful 
instrument which has multiphed astrononiical knowledge 
during the last half century, demonstrates, with mathe- 
matical certainty, the rotation' of the earth, and multiphes 
millionfold the certainty of the earth's revohition. 

Actual Motions of Stars. Before leaving this topic we 
should notice that other changes in the colors of stars show 
that some are actually approaching the earth at a uniform 
rate, and some are receding from it. Careful observa- 
tions at long intervals show other changes in the posi- 
tions of stars. The latter motion of a star is called its 



THE PARALLAX OF STARS 109 

proper motion to distinguish it from the apparent motion 
it has in common with other stars due to the motions of 
the earth. The spectroscope also assists in the demon- 
stration that the sun with the earth and the rest of the 
planets and their attendant satelUtes is moving rapidly 
toward the constellation Hercules. 

Elements of Orbit Determined by the Spectroscope. As 
an instance of the use of the spectroscope in determining 
motions of celestial bodies, we may cite the recent calcu- 
lations of Professor Kustner, Director of the Bonn Observa- 
tory. Extending from June 24, 1904, to January 15, 1905, 
he made careful observations and "photographs of the 
spectrographic Unes shown by Arcturus. He then made 
calculations based upon a microscopic examination of 
the photographic plates, and was able to determine (a) the 
size of the earth's orbit, (b) its form, (c) the rate of the 
earth's motion, and (d) the rate at which the solar system 
and Arcturus are approaching each other (10,849 miles per 
hour, though not in a direct hne). 

The Parallax of Stars. Since the days of Copernicus 
(1473-1543) the theory of the revolution of the earth 
around the sun has been very generally accepted. Tycho 
Brahe (1546-1601), however, and some other astrono- 
mers, rejected this theory because they argued that if the 
earth had a motion across the great distance claimed for its 
orbit, stars would change their positions in relation to the 
earth, and they could detect no such change. Little did 
iihey reahze the tremendous distances of the stars. It was 
not until 1838 that an astronomer succeeded in getting 
the orbital or heliocentric parallax of a star. The German 
astronomer Bessel then discovered that the faint star 61 
Cygni is annually displaced to the extent of 0.4". Since 
then about forty stars have been found to have measurable 



110 



THE EARTH'S REVOLUTION 



parallaxes, thus miiltiplying the proofs of the motion of 
the earth around the sun. 
Displacement of a Star Varies with its Distance. Figure 34 
_^ shows that the amount of 

the displacement of a star 
in the background of the 
heavens owing to a change 
in the position of the earth, 
varies with the distance of 
the star. The nearer the 
star, the greater the displace- 
ment; in every instance, how- 
ever, this apparent shifting of 
a star is exceedingly minute, 
owing to the great distance 
(see pp. 45, 246) of the very 
nearest of the stars. 

Since students often con- 
fuse the apparent orbit of a 
star described under aberra- 
tion of light with that due to the parallax, we may make 
the following comparisons: 




Fig- 34 



Aberrational Orbit 

1. The earth's rapid motion 
causes the rays of light to slant 
(apparently) into the telescope so 
that, as the earth changes its 
direction in going around the sun, 
the star seems to shift slightly 
about. 

2. This orbit has the same 
maximum width for all stars, 
however near or distant. 



Parallactic Orbit 

1. As the earth moves about 
in its orbit the stars seem to 
move about upon the back- 
ground of the celestial sphere. 



2. This orbit varies in width 
with the distance of the star; the 
nearer the star, the greater the 
width. 



winter constellations invisible in summer 111 

Effects of Earth's Revolution 

Winter Constellations Invisible in Summer. You have 
doubtless observed that some constellations which are 
visible on a winter's night cannot be seen on a summer's 
night. In January, the beautiful constellation Orion may 
be seen early in the evening and the whole night through; 
in July, not at- all. That this is due to the revolution of 
the earth around the sun may readily be made apparent. 
In the daytime we cannot easily see the stars around the 




sun, because of its great Hght and the peculiar properties 
of the atmosphere; six months from now the earth will 
have moved halfway aroimd the sun, and we shall be 
between the sun and the stars he now hides from view, 
and at night the stars now invisible will be visible. 

If you have made a record of the observations suggested 
in Chapter I, you will now find that Exhibit I (Fig. 35), 
shows that the Big Dipper and other star groups have 
sUghtly changed their relative positions for the same time 
of night, making a little more than one complete rotation 
during each twenty-four hours. In other words, the stars 



112 



THE EARTH'S REVOLUTION 



have been gaining a little on the sun in the apparent daily 
swing of the celestial sphere around the earth. 
The reasons for this may be understobd from a careful 




Fig. 36 



study of Figure 36. The outer circle, which should be 
indefinitely great, represents the celestial sphere; the iimer 
ellipse, the path of the earth around the sun. Now the sun 
does not seem to be, as it really is, relatively near the 
earth, but is projected into the celestial sphere among the 



TWO APPAKENT MOTIONS OF THE SUN 113 

stars. "V\Tien the earth is at point A the sun is seen among 
.the stars at a; when the earth has moved to B the sun 
seems to have moved to b, and so on throughout the annual 
orbit. The sun, therefore, seems to creep around the celestial 
sphere among the stars at the same rate and in the same 
direction as the earth moves in its orbit. If you walk around 
a room with someone standing in the center, you will see 
that his image may be projected upon the wall opposite, 
and as you walk around, his image on the wall will move 
around in the same direction. Thus the sun seems to move 
in the celestial sphere in the same direction and at the same 
rate as the earth moves around the sun. 

Two Apparent Motions of the Sun: Daily Westward, 
Annual Eastward. The sun, then, has two apparent 
motions, — a daily swing arotmd the earth with the celestial 
sphere, and this annual motion in the celestial sphere amotig 
the stars. The first motion is in a direction opposite to that 
of the earth's rotation and is from east to west, the second 
is in the same direction as the earth's revolution and is 
from west to east. If this is not readily seen from the 
foregoing statements and the diagram, think again of the 
rotation of the earth making an apparent rotation of the 
celestial sphere in the opposite direction, the reasons 
why the sun and moon seem to rise in the east and set in 
the west; then think of the motion of the earth around the 
sun by which the sun is projected among certain stars 
and then among other stars, seeming to creep among 
them from west to east. 

After seeing this clearly, think of yourself as facing the 
rising sun and a star which is also rising. Now imagine 
tl^e earth to have rotated once, a day to have elapsed, 
and the earth to have gone a day's journey in its orbit in 
the direction corresponding to upward. The sun would 

JO. MATH. GEO. — 8 



114 THE EAKTH'S REVOLUTION 

not then be on the horizon, but, the earth having moved 
" upward," it would be somewhat below the horizon. 
The same star, however, would be on the horizon, for the 
earth does not change its position in relation to the stars. 
After another rotation the earth wotild be, relative to the 
stars viewed in that direction, higher up in its orbit and 
the sun farther below the horizon when the star was just 
rising. In three months when the star rose the sun would 
be nearly beneath one's feet, or it would be midnight; in 
six months we should be on the other side of the sun, 
and it would be setting when the star was rising;' in nine 
months the earth would have covered the " downward " 
quadrant of its journey around the sun', and the star 
would rise at noon; twelve months later the sun and 
star would rise together again. If the sun and a star 
set together one evening, on the next evening the star 
would set a little before the sim, the next night earlier 
still. 

Since the sun passes around its orbit, 360°, in a year, 
365 days, it passes over a space of nearly one degree each 
day. The diameter of the sun as seen from the earth 
covers about half a degree of the celestial sphere. During 
one rotation of the earth, then, the sun creeps eastward 
among the stars about twice its own width. A star rising 
with the sun will gain on the sun nearly sis of a day 
during each rotation, or a little less than four minutes. 
The sun sets nearly four minutes later than the. star with 
which it set the day before. 

Sidereal Day. Solar Day. The time from star-rise to 
star-rise, or an exact rotation of the earth, is called a 
sidereal day. Its exact length is 23 h. 56 m. 4.09 s. The 
time between two successive passages of the sun over a 
given meridian, or from noon by the sun until the next 



CAUSES OP APPARENT MOTIONS OF THE SUN 115 

noon by the sun, is called a solar day* Its length varies 
somewhat, for reasons to be explained later, but aver- 
ages twenty-four hours. When we say " day," if it is not 
otherwise quahfied, we usually mean an average solar 
day divided into twenty-four hours, from midnight to 
midnight. The term "hour," too, when not otherwse 
qualified, refers to one twenty-fourth of a mean solar 
day. 

Causes of Apparent Motions of the Sun. The apparent 
motions of the sun are due to the real motions of the 




Fig. 37 



earth. If the earth moved slowly around the sun, the 
sun would appear to move slowly among the stars. Just 
as we know the direction and rate of the earth's rotation 
by observing the direction and rate of the apparent rota- 

* A solar day is sometimes defined as the interval from sunrise to 
sunrise again. This is true only at the equator. The length of the 
solar day corresponding to February 12, May 15, July 27, or November 
3, is almost exactly twenty-four hours. The time intervening between 
sunrise and sunrise again varies greatly with the latitude and season. 
On the dates named a solar day at the pole is twenty-four hours long, 
as it is everywhere else on earth. The time from sunrise to sunrise 
again, however, is almost six months at either pole. 



116 



THE EARTH'S REVOLUTION 



tion of the celestial sphere, we know the direction and 
rate of the earth's revolution by observing the direction 
and rate of the sun's apparent annual motion. 

The Ecliptic. The path which the center of the sun 
seems to trace aroimd the celestial sphere in its annual 




Fig. 38. Celestial sphere, showing zodiac 



orbit is called the ecliptic * The line traced by the center 
of the earth in its revolution about the sun is its orbit. 
Since the sun's apparent annual revolution around the sky 
is due to the earth's actual motion about the sun, the path 
of the sun, the ecliptic, must he in the same plane with the 

* So called because eclipses can occur only when the moon crosses 
the plane of the ecliptic. 



THE ZODIAC 117 

earth's orbit. The earth's equator and parallels, if ex- 
tended, would coincide with the celestial equator and 
parallels; similarly, the earth's orbit, if expanded in the 
same plane, would coincide with the ecliptic. We often 
use interchangeably the expressions " plane of the earth's 
orbit " and " plane of the ecliptic." 

The Zodiac. The orbits of the different planets and of 
the moon are inclined somewhat to the plane of the eclip- 
tic, but, excepting some of the minor planets, not more 
than eight degrees. The moon and principal planets, 
therefore, are never more than eight degrees from the 
pathway of the sun. This belt sixteen degrees wide, with 
the echptic as the center, is called the zodiac (more fully 
discussed in the Appendix, p. 293). Since the sun appears 
to pass arotmd the center of the zodiac once each year, 
the ancients, who observed these facts, divided it into 
twelve parts, one for each month, naming each part from 
some constellation in it. It is probably more nearly cor- 
rect historically, to say that these twelve constellations 
got their names originally from the position of the sun in 
the zodiac. Libra, the Balance, probably got its name 
from the fact that in ancient days the sun was among the 
group of stars thus named about September 23, when the 
days and nights are equal, thus balancing. In some such 
way these parts came to be called the " twelve signs of 
the zodiac," one for each month. 

The facts in this chapter concerning the apparent 
annual motion of the sun were well known to the ancients, 
possibly even. more generally than they are to-day. The 
reason for this is because there were few calendars and 
almanacs in the earher days of mankind, and people had 
to reckon their days by noting the position of the sun. 
Thus, instead of sa)dng that the date of his famous 



118 THE EARTH'S REVOLUTION 

journey to Canterbury was about the middle of April, 
Chaucer says it was 

• When Zephinis eek with his sweete breeth 
Enspired hath in every holt and heath 
The tendre croppes, and the younge sonne 
Hath in the Ram his halfe course yronne. 

Even if clothed in modern Enghsh such a desbription 
would be unintelhgible to a large proportion of the stu- 
dents of to-day, and would need some such translation as 
the following: 

" AVhen the west wind of spring with its sweet breath 
hath inspired or given new hfe in every field and heath 
to the tender crops, and the young sun (young because it 
had got only half way through the sign Aries, the Ram, 
which marked the beginning of the new year in Chaucer's 
day) hath run half his course through the sign the Ram." 

Obliquity of the Ecliptic. The orbit of the earth is 
not at right angles to the axis. If it were, the ecliptic 
would coincide with the celestial equator. The plane of 
the echptic and the plane of the celestial equator form an 
angle of nearly * 23^°. This is called the obhquity of the 
echptic. We sometimes speak of this as the inchnation of 
the earth's axis from a perpendicular to the plane of its 
orbit. 

Since the plane of the echptic forms an angle of 23^^° with 
the plane of the equator, the sun in its apparent annual 
course around in the echptic crosses the celestial equator 

* The exact amount varies slightly from year to year. The follow- 
ing table is taken from the Nautical Almanac, Newcomb's Calculations: 

1903 23° 27' 6.86" 1906 23° 27' 5.45" 

1904 23° 27' 6.39" 1907 23° 27' 4.98" 

1905 23° 27' 5.92" 1908 23° 27' 4.51" 



VARYING SPEED OP THE EARTH 



119 




Fig. 39 



twice each year, and at one season gets 23^° north of it, 
and at the opposite season 23^° south of it. The sun thus 
never gets nearer the pole of the celestial sphere than 66^°. 
On March 21 and September 23 the sun is on the celes- 
tial equator. On June 21 
and December 22 the sun 
is 23i° from the celestial 
equator. 

Earth's Orbit. We have 
learned that the earth's orbit 
is an elUpse, and the sim is 
at a focus of it. While the 
eccentricity is not great, and 
when reduced in scale the 
orbit does not differ materi- 
ally from a circle, the differ- 
ence is sufficient to make an appreciable difference in the 
rate of the earth's motion in different parts of its orbit. 
Figure 113, p. 285, represents the orbit of the earth, greatly 
exaggerating the ellipticity. The point in the orbit nearest 
the sun is called perihehon (from peri, aroimd or near, and 
helios, the sun). This point is about 91^ million miles from 
the sun, and the earth reaches it about December 31st. 
The point in the earth's orbit farthest from the sun is 
called aphelion (from a, away from, and helios, sim). Its 
distance is about 94^ miUion miles, and the earth reaches 
it about July 1st. 

Varying Speed of the Earth. According to the law of 
gravitation, the earth moves faster in its orbit when near 
perihelion, and slower when near apheUon. In December 
and January the earth moves fastest in its orbit, and 
during that period the sim moves fastest in the ecliptic 
and falls farther behind the stars in their rotation in the 



120 



THE EARTH'S REVOLUTION 



celestial sphere. Solar days are thus longer then than 
they are in midsummer when the earth moves more slowly 
in its orbit and more nearly keeps up with the stars. 

Imagine the sun and a star are rising together January 
1st. After one exact rotation of the earth, a sidereal day, 
the star will be rising again, but since the earth has moved 
rapidly in its course around the sun, the sun is somewhat 
farther behind the star than it would be in summer when 
the earth moved more slowly around the sun. At star- 
rise January 3d, the sun is behind still farther, and in 
the course of a few weeks the sun will be several minutes 
behind the point where it would be if the earth's orbital 
motion were uniform. The sun is then said to be slow of 
the average sun. In July the sun creeps back less rapidly 
in the ecliptic, and thus a solar day is more nearly the 
same length as a sidereal day, and hence longer than the 
average. 

Another factor modifies the foregoing statements. The 
daily courses of the stars swinging around with the celes- 
tial sphere are parallel and are at right angles to the axis. 

The sun in its annual 

26° 

25° 

24° 

'23° 
2/0 



path creeps diagonally 
across their courses. 
When farthest from the 
celestial equator, in June 
and in December, the 
sun's movement in the 
ecliptic is nearly parallel 
Fig. 40 to the courses of the 

stars (Fig. 40); as it 
gets nearer the celestial equator, in March and in Sep- 
tember, the course is more obhque. Hence in the latter 
part of June and of December, the sun, creeping back in 



e ^, 



SIDEREAL DAY SHOKTEK THAN SOLAR DAY 121 



the ecliptic, falls farther behind the stars and becomes 
slower than the average. In the latter part of March and 
of September the sun creeps in a more diagonal course and 
hence does not fall so far behind the stars in going the 
same distance, and thus becomes faster than the average 
(Fig. 41). 

Some solar days being longer than others, and the 
Sim being sometimes 



slow and sometimes 
fast, together with 
standard time adop- 
tions whereby most 
places have their 
watches set by mean 
solar time at some 
given meridian, make 



3° 

0° 
2° 



Fig. 41 

it unsafe to set one's 

watch by the sun without making many corrections. 

The shortest day in the northern hemisphere is about 
December 22d; about that time the sun is neither fast nor 
slow, but it then begins to get slow. So as the days get 
longer the sun does not rise any earher until about the 
second week of January. After Christmas one may notice 
the later and later time of sunsets. In schools in the 
northern states beginning work at 8 o'clock in the morning, 
it is noticed that the mornings are actually darker for a 
while after the Christmas hoUdays than before, though the 
shortest day of the year has passed. 

Sidereal Day Shorter than Solar Day. If one wanted 
to set his watch by the stars, he would be obhged to 
remember that sidereal days are shorter than solar days; 
if the star observed is in a certain position at a given time 
of night, it will be there nearly four minutes earlier the 



122 THE EAETH'S^ REVOLUTION 

next evening. The Greek dramatist Euripides (480-407 
B.C.), in his tragedy " Rhesus," makes the Chorus say: 

Whose is the ■ guard? Who takes my turn? The first signs 
are setting, and the seven Pleiades are in the sky, and the Eagle 
glides midway through the sky. Awake! See ye not the brilliancy 
of the moon? Morn, mom, indeed is approaching, and hither is one 
of the forewarning stars. 



Summary, 
Note carefully these propositions: 

1. The earth's orbit is an ellipse. 

2. The earth's orbital direction is the same as the direction of its 

axial motion. 

3. The rate of the earth's rotation is uniform, hence sidereal days 

are of equal length. 

4. The orbit of the earth is in nearly the samo plane as that of the 

equator. 

5. The earth's revolution around the sun makes the sun seem to 

creep backward among the stars from west to east, falling 
behind them about a degree a day. The stars seem to swing 
around the earth, daily gaining about four minutes upon the 
sun. 

6. The rate of the earth's orbital motion determines the rate of the 

sun's apparent annual backward motion among the stars. 

7. The rate of the earth's orbital motion varies, being fastest when 

the earth is nearest the sun or in perihelion, and slowest when 
farthest from the sun or in aphelion.. 

8. The sun's apparent annual motion, backward or eastward among 

the stars, is greater when in or near perihelion (December 31) 
than at any other time. 

9. The length of solar days varies, averaging 24 hours in length. 

There are two reasons for this variation. 
a. Because the earth's orbital motion is not uniform, it being faster 

when nearer the sun, and slower when farther from it. 
6. Because when near the equinoxes the apparent annual motion 

of the sun in the celestial sphere is more diagonal than when 

near the tropics. 



SUN PAST OR SUN SLOW 123 

10. Because of these two sets of causes, solar days are more than 24 
hours in length from December 25 to April 15 and from June 
15 to September 1, and less than 24 hours in length from April 
15 to June 15 and from September 1 to December 25. 

Equation of Time 

Sun Fast or Sun Slow. The relation of the apparent 
solar time to mean solar time is called the equation of 
time. As just shown, the apparent eastward motion of 
the sun in the ecliptic is faster than the average twice a 
year, and slower than the average twice a year. A ficti- 
tious sun is imagined to move at a uniform rate eastward 
in the celestial equator, starting with the apparent sun at 
the vernal equinox (see Equinox in Glossary) and com- 
pleting its annual course around the celestial sphere in 
the same time in which the sun apparently makes its cir- 
cuit of the ecliptic. AVhile , excepting four times a year, 
the apparent sun is fast or slow as compared with this 
fictitious sun which indicates mean solar time, their differ- 
ence at any moment, or the equation of time, may be 
accurately calculated. 

The equation of time is indicated in various ways. The 
usual method is to indicate the time by which the appar- 
ent sun is faster than the average by a minus sign, and the 
time by which it is slower than the average by a plus sign. 
The apparent time and the equation of time thus indicated, 
when combined, will give the mean time. Thus, if the sun 
indicates noon (apparent time), and we know the equation 
to be — 7 m. (sun fast, 7 m.), we know it is 11 h. 53 m., a.m. 
by mean solar time. 

Any almanac shows the equation of time for any day of 
the year. It is indicated in a variety of ways. 

a. In the World Almanac it is given imder the title 



124 THE EARTH'S REVOLUTION 

" Sun on' Meridian." The local mean solar time of the 
sun's crossing a meridian is given to the nearest second. 
Thus Jan. 1, 1908, it is given as 12 h. 3 m. 16 s. We know 
from this that the apparent sun is 3 m. 16 s. slow of the 
average on that date. 

b. In the Old Farmer's Almanac the equation of time is 
given in a column headed " Sun Fast," or " Sun Slow." 

c. In some places the equation of time is indicated by 
the words, " clock ahead of sun," and " clock behind sun." 
Of course the student knows from this that if the clock is 
ahead of the sun, the sun is slower than the average, and, 
conversely, if the clock is behind the sun, the latter must 
be faster than the average. 

d. Most almanacs give times of sunrise and of sunset. 
Now half way between sunrise and sunset it is apparent 
noon. Suppose the sun rises at 7:24 o'clock, a.m., and 
sets at 4:43 o'clock, p.m. Half way between those times 
is 12:03^ o'clock, the time when the sun is on the 
meridian, and thus the sun is 3^ minutes slow (Jan. 1, at 
New York). 

e. The Nautical Almanac * has the most detailed and 
accurate data obtainable. Table II for each month gives 
in the column " Equation of Time " the number of min- 
utes and seconds to be added to or subtracted from 12 
o'clock noon at Greenwich for the apparent sun time. The 
adjoining column gives the difference for one hour to be 
added when the sun is gaining, or subtracted when the sun 
is losing, for places east of Greenwich, and vice versa for 
places west. 

Whether or not the student has access to a copy of the 

* Prepared annually three years in advance, by the Professor of 
Mathematics, United States Navy, Washington, D. C. It is sold by 
the Bureau of Equipment at actual cost of publication, one dollar. 



SU]S[ FAST OR SUN SLOW 



125 



Nautical Almanac it may be of interest to notice the use 
of this table. 



AT GKBENWICH MEAN NOON. 





4 
1 

1 
o 


THE 


SUN'S 




Equation 
of Time 
to be 
Sub- 
tracted 
from 
Mean 
Time 


Dife. 

for 

1 

Hour 


Sidereal 


1 

•s 


Apparent 

Right 
Ascension 


DifE. 

for 

IHour 


Apparent 
Declination 


Diff. 

for 

IHonr 


or Right 

Ascension of 

Mean Sun 


Wed. 
Thur. 
Frid. 


1 

2 
3 


h m s 
18 42 9.88 
18 46 35.09 
18 50 59.99 


11.057 
11.044 
11.030 


o / 

S. 23 5 

23 1 

22 66 


It 

47.3 
6.3 

57.7 


// 
+ 11.13 
12.28 
13.42 


m s 
3 10.29 

3 38.93 

4 7.28 


1.200 
1.188 
1.174 


h m 8 
18 38 59.60 
18 42 66.16 
18 46 52.71 


Sat. 

SUN. 

Mon. 


4 
5 
6 


18 55 24.54 

18 69 48.70 

19 4 12.46 


11.015 
10.998 
10.979 


22 50 
22 44 
22 37 


21.8 
18.6 
48.2 


+ 14.56 
15.70 
16.82 


4 35.27 

5 2.87 
5 30.06 


1.158 
1.141 
1.123 


18 50 49.27 
18 54 45.83 
18 68 42.39 


Tues. 
Wed. 
Thur. 


7 
8 
9 


19 8 35.74 
19 12 58.56 
19 17 20.85 


10.969 
10.939 
10.918 


22 30 
22 23 
22 15 


51.0 
27.1 
36.8 


+ 17.94 
19.04 
20.14 


6 56.80 
6 23.06 
6 48.79 


1.104 
1.083 
1.061 


19 2 38.94 
19 6 35.60 
19 10 32.06 


Frid. 

Sat. 

SUN. 


10 
11 
12 


19 21 42.61 
19 26 3.79 
19 30 24.39 


10.895 
10.871 
10.846 


22 7 
21 58 
21 49 


20.2 
37.7 
29.5 


+ 21.23 
22.30 
23.37 


7 13.99 

7 38.62 

8 2.66 


1.038 
1.014 
0.989 


19 14 28.82 
19 18 26.17 
19 22 21.73 



Part of a page from The American EpheTtleris and Nautical Almanac, Jan. 1908. 

This table indicates that at 12 o'clock noon, on the 
meridian of Greenwich on Jan. 1, 1908, the sun is slow 3 m. 
10.29 s., and is losing 1.200 s. each hour from that moment. 
We know it is losing, for we find that on January 2 the 
sun is slow 3 m. 38.93 s., and by that time its rate of loss 
is slightly less, being 1.188 s. each hour. 

Suppose you are at Hamburg on Jan. 1, 1908, when 
it is noon according to standard time of Germany, one 
hour before Greenwich mean noon. The equation of time 
will be the same as at Greenwich less 1.200 s. for the hour's 
difference, or (3 m. 10.29 s. - 1 .200 s. ) 3 m. 9.09 s. If you are 
at New York on that date and it is noon, Eastern standard 



126 THE EARTH'S REVOLUTION 

time, five hours after Greenwich noon, it is obvious that 
the sun is 5 X 1.200 s. or 6 s. slower than it was at Green- 
wich mean noon. The equation of time at New York 
would then be 3 m. 10.29 s. + 6 s. or 3 m. 16.29 s. ' 

/. The Analemma graphically indicates the approximate 
equation of time for any day of the year, and also indicates 
the dechnation of the sun (or its distance from the celestial 
equator). Since our year has 365f days, the equation of 
time for a given date of one year will not be quite the 
same as that of the. same date in a succeeding year. That 
for 1910 will be approximately one fourth of a day or six 
hours later in each day than for 1909; that is, the table 
for Greenwich in 1910 will be very nearly correct for Cen- 
tral United States in 1909. Since for the ordinary pur- 
poses of the student using this book an error of a few 
seconds is inappreciable, the analemma will answer for 
most of his calculations. 

The vertical lines of the analemma represent the num- 
ber of minutes the apparent sun is slow or fast as com- 
pared with the mean sun. For example, the dot repre- 
senting February 25 is a little over half way between the 
hnes representing sun slow 12 m. and 14 m. The sun is then 
slow about 13 m. 18 s. It will be observed that April 15, 
June 15, September 1, and December 25 are on the central 
line. The equation of time is then zero, and the sun may 
be said to be " on time." Persons hving in the United 
States on the 90th meridian will see the shadow due north 
at 12 o'clock on those days; if west of a standard time 
meridian one will note the north shadow when it is past 
12 o'clock, four minutes for every degree; and, if east of 
a standard time meridian, before 12 o'clock four minutes 
for each degree. Since the analemma shows how fast or 
slow the sun is each day, it is obvious that, knowing one's 



SUN FAST OR SUN SLOW 



127 




128 THE EARTH'S REVOLUTION 

longitude, one can set his watch by the sun by reference 
to this diagram, or, having correct clock time, one can 
ascertain his longitude. 

Uses of the Analemma 

To Ascertain Your Longitude. To do this your watch 
must show correct standard time. You must also have a 
true north-south line. 

1. CarefuUy observe the time when the shadow is north. 
Ascertain from the analemma the number of minutes and 
seconds the sun is fast or slow. 

2. If fast, add that amount to the time by your watch ; 
if slow, subtract. This gives your mean local time. 

3. Divide the minutes and seconds past or before twelve 
by four. This gives you the number of degrees and 
minutes you are from the standard time meridian. If 
the corrected time is before twelve, you are east of it; if 
after, you are west of it. 

4. Subtract (or add) the number of degrees you are 
east (or west) of the standard time meridian, and this is 
your longitude. 

For example, say the date is October 5th. 1. Your 
watch says 12 h. 10 m. 30 s., p.m., when the shadow is north. 
The analemma shows the sun to be 11 m. 30 s. fast. 2. The 
sun being fast, you add these and get 12:22 o'clock, p.m. 
This is the mean local time of your place. 3. Dividing 
the minutes past twelve by four, you get 5 m. 30 s. This is 
the number of degrees and minutes you are west from the 
standard meridian. If you live in the Central standard 
time belt of the United States, your longitude is 90° plus 
5° 30', or 95° 30'. If you are in the Eastern time belt, 
it is 75° plus 5° 30'. If you are in Spain, it is 0° plus 
5° 30', and so on. 



TO SET YOUR WATCH 129 

To Set Your Watch. To do this you must know your 
longitude and have a true north-south Une. 

1. Find the difference between your longitude and that 
of the standard time meridian in accordance with which 
you wish to set your watch. In Eastern United States the 
standard time meridian is the 75th, in Central United 
States the 90th, etc. 

2. Multiply the number of degrees and seconds of 
difference by four. This gives you the number of minutes 
and seconds your time is faster or slower than local time. 
If you are east of the standard meridian, your watch must 
be set slower than local time ; if west, faster. 

3. From the analenama observe the position of the sun 
whether fast or slow and how much. If fast, subtract 
that time from the time obtained in step two; if slow, add. 
This gives you the time before or after twelve when the 
shadow will be north; before twelve if you are east of the 
standard time meridian, after twelve if you are west. 

4. Carefully set your watch at the time indicated in step 
three when the sun's shadow crosses the north-south line. 

For example, suppose your longitude is 87° 37' W. 
(Chicago). 1. The difference between your longitude and 
your standard time meridian, 90°, is 2° 23'. 2. Multi- 
plying this difference by four we get 9° 32', the minutes 
and seconds your time is slower than the sun's average 
time. That is, the sun on the average casts a north 
shadow at 11 h. 50 m. 28 s. at your longitude. 3. From the 
analemma we see the sun is 14 m. 15 s. slow on February 6. 
The time being slow, we add this to 11 h. 50 m. 28 s. and 
get 12 h. 4 m. 43 s., or 4 m. 43 s. past twelve when the 
shadow will be north. 4. Just before the shadow is north 
get your watch ready, and the moment the shadow is north 
set it 4 m. 43 s. past twelve. 

JO. MATH. GEO. — 9 



130 THE EARTH'S REVOLUTION 

To Strike a North-South Line. To do this you must 
know your longitude and have correct time. 

Steps 1, 2, and 3 are exactly as in the foregoing explana- 
tion how to set your watch by the sun. At the time you 
obtain in step 3 you know the shadow is north; then draw 
the Une of the shadow, or, if out of doors, drive stakes or 
otherwise indicate the line of the shadow. 

To Ascertain Your Latitude. This use of the analemma 
is reserved for later discussion. 

Civil and Astronomical Days. The mean solar day of 
twenty-four hours reckoned from midnight is called a civil 
day, and among all Christian nations has the sanction of 
law and usage. Since astronomers work at night they 
reckon a day from noon. Thus the civil forenoon is 
dated a day ahead of the astronomical day, the afternoon 
being the last half of the civil day but the beginning of 
the astronomical day. Before the invention of clocks and 
watches, the simdial was the common standard for the 
time during each day, and this, as we have seen, is a con- 
stantly varying one. When clocks were invented it was 
found impossible to have them so adjusted as to gain or 
lose with the sun. Until 1815 a civil day in France was a 
day according to the actual position of the stm, and hence 
was a very uncertain affair. 

A Few Facts : Do You Understand Them ? 

1. A day of twenty-four hours as we commonly use the 
term, is not one rotation of the earth. A solar day is a 
little more than one complete rotation and averages 
exactly twenty-four hours in length. This is a civil or 
legal day. 

2. A sidereal day is the time of one rotation of the 
earth on its axis. 



CIVIL AND ASTRONOMICAL DAYS 131 

3. There are 366 rotations of the earth (sidereal days) 
in one year of 365 days (solar days). 

4. A sundial records apparent or actual sun time, which 
is the same as mean sun time only four times a year. 

5. A clock records mean sun time, and thus corresponds 
to sundial time only four times a year. 

6. In many cities using standard time the shadow of the 
sun is never in a north-south line when the cIock strikes 
twelve. This is true of aU cities more than 4° east or 
west of the meridian on which their standard time is based. 

7. Any city within 4° of its standard time meridian will 
have north-south shadow Unes at twelve o'clock no more 
than four times a year at the most. Strictly speaking, 
practically no city ever has a shadow exactly north-south 
at twelve o'clock. 



CHAPTER VII 

TIME AND THE CALENDAR 

" In the early days of mankind, it is not probable that 
there was any concern at all about dates, or seasons, or 
years. Herodotus is called the father of history, and his 
history does not contain a single date. Substantially 
the same may be said of Thucydides, who wrote only a 
little later — somewhat over 400 B.C. If Geography and 
Chronology are the two eyes of history, then some histories 
are bhnd of the one eye and can see but little out of the 
other." * 

Sidereal Year. Tropical Year. As there are two kinds 
of days, solar and sidereal, there are two kinds of years, 
solar or tropical years and sidereal years, but for very 
different reasons. The sidereal year is the time elapsing 
between the passage of the earth's center over a given 
point in its orbit until it crosses it again. For reasons 
not properly discussed here (see Precession of the Equi- 
noxes, p. 286), the point in the orbit where the earth is 
when the vertical ray is on the equator shifts slightly 
westward so that we reach the point of the vernal equinox 
a second time a few minutes before a sidereal year has 
elapsed. The time elapsing from the sun's crossing of the 
celestial equator in the spring until the crossing the next 
spring is a tropical year and is what we mean when we 
say " a year." \ Since it is the tropical year that we 

* R W. Farland in Popular Astronomy for February, 1895. 

t A third kind of year is considered in astronomy, the anomalistic 
year, the time occupied by the earth in traveling from perihelion to 
perihelion again. Its length is 365 d. 6 h. 13 m, 48,098. The lunar year, 

182 



THE MOON THE MEASURER 133 

attempt to fit into an annual calendar and which marks 
the year of seasons, it is well to remember its length: 
365 d. 5 h. 48 m. 45.51 s. (365.2422 d.). The adjust- 
ment of the days, weeks, and months into a calendar 
that does not change from year to year but brings 
the annual hohdays around in the proper seasons, has 
been a difficult task -for the human race to accomplish. 
If the length of the year were an even number of days 
and that number was- exactly divisible by twelve, 
seven, and four, we could easily have seven days in a week, 
four weeks in a month, and twelve months in a year and 
have po time to carry over into another year or month. 

The Moon the Measurer. Among the ancients the 
moon was the great measurer of time, our word month 
comes from the word moon, and in connection with its 
changing phases religious feasts and celebrations were 
observed. Even to-day we reckon Easter and some other 
holy days by reference to the moon. Now the natural 
units of time are the solar day, the lunar month (about 
29J days), and the tropical year. But their lengths are 
prime to each other. For some reasons not clearly known, 
but believed to be in accordance with the four phases of 
the moon, the ancient Egyptians and Chaldeans divided 
the month into four weeks of seven days each. The 
addition of the week as a unit of time which is naturally 
related only to the day, made confusion worse confounded. 
Various devices have been used at different times to make 
the same date come around regularly in the same season 
year after year, but changes made by priests who were 
ignorant as to the astronomical data and by more igno- 
rant kings often resulted in great confusion. The very 

twelve new moons, is about eleven days shorter than the tropical 
year. The length of a sidereal year is 365 d. 6 h. 9 m. 8.97 s. 



134 TIME AND THE CALENDAR 

exact length of the solar year in the possession of the 
ancient Egyptians seems to have been little regarded. 

Early Roman Calendar. Since our calendar is the same 
as that worked out by the Romans, a brief sketch of their 
system may be helpful. The ancient Romans seem to 
have had ten months, the first being March. We can see 
that this was the case from the fact that September means 
seventh; October, eighth; November, ninth; and Decem- 
ber, tenth. It was possibly during the reign of Numa 
that two months were added, January and February. 
There are about 29^ days in a lunar month, or from one 
new moon to the next, so to have their months conform 
to the moons they were given 29 and 30 days alter- 
nately, beginning with January. This gave them twelve 
lunar months in a year of 354 days. It was thought 
unlucky to have the number even, so a day was added for 
luck. 

This year, having but 355 days, was over ten days too 
short, so -festivals that came in the summer season would 
appear ten days earlier each year, until those dedicated 
to Bacchus, the god of wine, came when the grapes were 
still green, and those of Ceres, the goddess of the harvest, 
before the heads of the wheat had appeared. To correct 
this an extra month was added, called Mercedonius, every 
second year. Since the length of this month was not fixed 
by law but was determined by the pontiffs, it gave rise 
to serious corruption and fraud, interfering with the collec- 
tion of debts by the dropping out of certain e5cpected 
dates, lengthening the terms of office of favorites, etc. 

The Julian and the Augustan Calendars. In the year 
46 B.C., Julius Caesar, aided by the Egyptian astronomer, 
Sosigenes, reformed the calendar. He decreed that begin- 
ning with January the months should have alternately 31 



Julian Augustan 


Jan. 


31 


31 


Feb. 


29-30 


28-29 


Mar. 


31 


31 


Apr. 


30 


30 


May 


31 


31 


June 


30 


30 


July 


31 


31 


Aug. 


30 


31 


Sept. 


31 


30 


Oct. 


30 


31 


Nov. 


31 


30 


Dec. 


30 


31 



THE GREGORIAN CALENDAR 135 

and 30 days, save February, to which was assigned 29 
days, and every fourth year an additional day. This 
made a year of exactly 365i days. Since the true year has 
365 days, 5 hours, 48 min., 45.51 sec, and the Julian year 
had 365 days, 6 hours, it was 11 
min., 14.49 sec. too long. 

During the reign of Augustus 
another day was taken from Feb- 
ruary and added to August in order 
that that month, the name of which 
had been changed from Sextilis to 
August in his honor, might have as 
many days in it as the month 
Quintihs, whose name had been 
changed to July in honor of Julius Caesar. To prevent the 
three months, July, August, and September, from having 31 
days each, such an arrangement being considered unlucky, 
Augustus ordered that one day be taken from September 
and added to October, one from November and added to 
December. Thus we find the easy plan of remembering 
the months having 31 days, every other one, was dis- 
arranged, and we must now count our knuckles or learn : 

"Thirty days hath September, April, June, and November. 
All the rest have thirty-one, save the second one alone, 
Which has four and twenty-four, till leap year gives it one day more." 

The Gregorian Calendar. This Juhan calendar, as it 
is called, was adopted by European countries just as they 
adopted other Roman customs. Its length was 365.25 
days, whereas the true length of the year is 365.2422 
days. While the error was only .0078 of a year, in the 
course of centuries this addition to the true year began 
to amount to days. By 1582 the difference had amounted 
to about 13 days, so that the time of the spring equinox, 



136 TIME AND THE CALENDAR 

when the sun crosses the celestial equator, occurred the 
11th of March. In that year Pope Gregory XIII reformed 
the calendar so that the March equinox might occur on 
March 21st, the same date as it did in the year 325 a.d., 
when the great Council of Nicsea was held which finally 
decided the method of reckoning Easter. One thousand 
two hundred and fifty-seven years had elapsed, each being 
11 min. 14 sec. too long. The error of 10 days was 
corrected by having the date following October 4th of that 
year recorded as October 15. To prevent a recurrence of 
the error, the Pope further decreed that thereafter the 
centurial years not divisible by 400 should not be counted 
as leap years. Thus the years 1600, 2000, 2400, etc., are 
leap years, but the years 1700, 1900, 2100, etc., are not 
leap years. This calculation reduces the error to a very 
low point, as according to the Gregorian calendar nearly 
4000 years must elapse before the error amounts to a 
single day. 

The Gregorian calendar was soon adopted in aU Roman 
Catholic countries, France recording the date after Decem- 
ber 9th as December 20th. It was adopted by Poland in 
1786, and by Hungary in 1787. Protestant Germany, 
Denmark, and Holland adopted it in 1700 and Protestant 
Switzerland in 1701. The Greek Catholic countries have 
not yet adopted this calendar and are now thirteen days 
behind our dates. Non-Christian countries ha've calendars 
of their own. 

In England and her colonies the change to the Gre- 
gorian system was effected in 1752 by having the date 
following September 2d read September 14. The change 
was violently opposed by some who seemed to think that 
changing the number assigned to a particular day modi- 
fied time itself, and the members of the Government are 



OLD STYLE AND NEW STYLE 



137 



SEPTEMBER. IX Month. 



Shall Fruits, -which none, but brulal E/es ftirvey, 
Unlouch'd grow ripe, jtilaftfd drop away 
Shall here Ih" irraiional, the/alvage Kind 
Lord ir o'er Stores by Heav'n for Man defign'd. 
And irample what mild Suns benignly raife. 
While Man mud loTe ihe Ufe. and Hfav'n the Praife » 
Shall it then be.'*" (Indignant here Hie rofe. 
Indignant, yet humane, iter Bofom glows) 

' No 



Remark. dayi.o'f. Grifj Qfetj l)pl.| Afpefls, y^ 



5 46 
5 47 
5 49 
5 !« 
S i 
5 M 
5 i + 



said to have been mobbed in London by laborers who 
cried " give us back our eleven days." 

Old Style and New Style. Dates of events occurring 
before this change 
are usually kept as 
they were then writ- 
ten, the letters o.s. 
sometimes being 
written after the 
date to signify the 
old style of dating. 
To translate a date 
into the Gregorian or 
new style, one must 
note the century in 
which it occurred. 
For example, Colum- 
bus discovered land 
Oct. 12, 1492, o.s. 
According to the 
Gregorian xjalendar 
a change of 10 days 
was necessary in 
1582. In 1500, leap 
year was coimted by 
the old style but 
should not have 
been counted by the 
new style. Hence, 
in the century end- 
ing 1 500, only 9 days 
difference had been made. So the discovery of America 
occurred October 12, o.s. or October 21, n.s. English 



mnd 1 , 666 
London burnt, ^ 
ond cloudt 
lui/b 
Day break 42^. 
I y part Trin. 

rain. 
Nait\. V .M«ft V 

fhen etiat 
St Matthew 
Days deer. J a6 

and 
16 paftTrin. 
Holy Rood, 

lAlindf, 

Ember Week. 
Plia/anr 
St. Michael, 



59 
57 
56 
5 55 
5 53 
5 S» 
5 51 
5 49 



TX\ 1 9j7>f tod obfipng 

t 17 'i 'iff 8 +0 
ijjliwithlj 7m- 
i( h fet 10 20 ftr 
Vri(e 1 1 51 

IS t^trmgrf 
difobJi^mo 

iOi ofiif. 

Hold yOut 
Coltmil 
Omii -r)' 9 5 
be/ore Dinner ; 
7*s>ife 8 o 
the fJl Bill f 
battt Tljinhng 
i t 1} at luflt 
ttl ^/hnp. 



•5- 7 



'3 

M 

X 7 



r 3 
16 

'9 
81) 

27 

n 1 1 

°z 9 



twl* * ^ y 



t..<i-.i D,,olJ,p„mt„ m, 61J Co«M. of StlTion .„d E«h„o,. 
.,«. ,'""'""'""./•"'. •"- M.,., .,.f„,cf,M, ,„a.|i c„„„ 

hr.r„.,'A,. .; Aa '• ^''"'' D.,, in,,, ,h,„ ,,,. f.„e would 
?"" '"PP'i'M. •"Ord/Ag to ih, ■NommM Djyi of the full) H>» 

tonii.rf ihfteof in toy wii't 
t.le. ■ " 



lA, i)i,o{ ,0 ihij Act conurned to the 
Ir notwtihfldnding. 
'c<.i.iA PlV,'!;,"'''!.'',"'"..'^''''?""' P'«f"ipllonl, .nd U. 

— tt 



Fig. 43. 



Page from Franklin's Almanac Sbowing 
Omission of Eleven Days, 1752. 



138 TIME AND THE CALENDAR 

12 
historians often write such dates October — , the upper date 

referring to old style and the lower to new style. 

A historian usually follows the dates in the calendar used 
by his country at the time of the event. If, however, the 
event refers to two nations having different calendars, both 
dates are given. Thus, throughout Macaulay's "History 
of England" one sees such dates as the following: Avaux, 

, ^ „ ,1689. (Vol. III.) A few dates in American his- 
Aug. 6 

tory prior to September, 1752, have been changed to agree 

with the new style. Thus Washington was born Feb. 11, 

1731, O.S., but we always write it Feb. 22, 1732. The 

reason why aU such dates are not translated into new 

style is because great confusion would result, and, besides, 

some incongruities would obtain. Thus the principal ship 

of Colimibus was wrecked Dec. 25, 1492, and Sir Isaac 

Newton was born Dec. 25, 1642, and since in each case 

this was Christmas, it would hardly do to record them as 

Christmas, Jan. 3, 1493, in the former instance, or as 

Christmas, Jan. 4, 1643, in the latter case, as we should 

have to do to write them in new style. 

The Beginning of the Year. With the ancient Romans 

the year had commenced with the March equinox, as 

we notice in the names of the last months, September, 

October, November, December, meaning 7th, 8th, 9th, 

10th, which could only have those names by coimting 

back to March as the first month. By the time of Julius 

Caesar the December solstice was commonly regarded as 

the beginning of the year, and he confirmed the change, - 

making his new year begin January first. The later 

Teutonic nations for a long time continued counting the 

beginning of the year from March 25th. In 1563, by an 



OLD STYLE IS STILL USED IN ENGLAND 139 

edict of Charles IX, France changed the time of the begin- 
ning of the year to January first. In 1600 Scotland made 
the same change and England did the same in 1752 when 
the Gregorian system was adopted there. Dates between 
the first of January and the twenty-fifth of March, from 
1600 to 1752 are in one year in Scotland and another year 
in England. In Macaulay's " History of England" (Vol. 
Ill, p. 258), he gives the following reference: Act. Pari. 
Scot., Mar. 19, 1689-90." The date being between Jan- 
uary 1st and March 25th in the interval between 1600 and 
1752, it was recorded as the year 1689 in England, and a 
year later, or 1690, in Scotland — Scotland dating the new 
year from January 1st, England from March 25th. This 
explains also why Washington's birthday was in 1731, o.s., 
and 1732, n.s., since English colonies used the same 
system of dating as the mother country. 

Old Style is still used in England's Treasury Depart- 
ment. " The old style is still retained in the accoimts 
of Her Majesty's Treasury. This is why the Christmas 
dividends are not considered due until Twelfth Day, and 
the midsummer dividends not till the 5th of July, and in 
just the same way it is not until the 5th of April that 
Lady Day is supposed to arrive. There is another piece 
of antiquity in the pubhc accounts. In the old times, the 
year was held to begin on the 25th of March, and this 
change is also still observed in the computations over 
which the Chancellor of the Exchequer presides. The 
consequence is, that the first day of the financial year is 
the 5th of April, being old Lady Day, and with that day 
the reckonings of our annual budgets begin and end." — 
London Times* Feb. 16, 1861. 

* Under the date of September 10, 1906, the same authority says 
that the facts above quoted obtain in England at the present time. 



140 TIME AND THE CALENDAR 

Greek Catholic Countries Use Old Style. The Greek 
Catholic countries, Russia, some of the Balkan states and 
Greece, still employ the old Julian calendar which now, 
with their counting 1900 as a leap year and our not 
counting it so, makes their dates 13 days behind ours. 
Dates in these countries recorded by Protestants or Roman 
Catholics or written for general circulation are commonly 
recorded in both styles by placing the Gregorian date 
under the Julian date. For example, the date we cele- 
brate as our national holiday would be written by an 

American in Russia as -r— The day we commem- 

July i ^ 

12 

orate as the anniversary of the birth of Christ, Dec. — ; the 

25 

day they commemorate, ~ — '■ — _' ^^^ - It should be 

Jan. 7, 1907 

remembered that if the date is before 1900 the differ- 
ence will be less than thirteen days. Steps are being 
taken in Russia looking to an early revision of the 
calendar. 

Mohammedan and Jewish Calendars. The old system 
employed before the time of the Caesars is still used by 
the Mohammedans and the Jews. The year of the former 
is the lunar year of 354^| days, and being about .03 of 
a year too short to correspond with the solar year, the 
same date passes through all seasons of the year in the 
course of 33 years. Their calendar dates from the year 
of the Hegira, or the flight of Mohammed, which occurred 
July, 622 A.D. If their year was a full solar year, their 
date corresponding to 1900 would be 622 years less than 
that number, or 1278, but being shorter in length there are 
more of them, and they write the date 1318, that year 
beginning with what to us was May 1. That is to say, 



CHALDEAN CALENDAR 141 

what we called May 1, 1900, they called the first day of 
their first month, Muharram, 1318. 

Chinese Calendar. The Chinese also use a lunar calen- 
dar; that is, with months based upon the phases of the 
moon, each month beginning with a new moon. Their 
months consequently have 29 and 30 days alternately. 
To correct the error due to so short a year, seven out of 
every nineteen years have thirteen months each. This 
still leaves the average year too short, so in every cycle of 
sixty years, twenty-two extra months are intercalated. 

Ancient Mexican Calendar. The ancient Mexicans had a 
calendar of 18 months of 20 days each and five additional 
days, with every fourth year a leap year. Their year began 
with the vernal equinox. 

Chaldean Calendar. Perhaps the most ancient calendar 
of which we have record, and the one which with modifi- 
cations became the basis of the Roman calendar which we 
have seen was handed down through successive genera- 
tions to us, was the calendar of the Chaldeans. Long 
before Abraham left Ur of the Chaldees (see Genesis xi, 
31; Nehemiah ix, 7, etc.) that city had a royal observatory, 
and Chaldeans had made subdivisions of the celestial sphere 
and worked out the calendar upon which ours is based. 

Few of us can fail to recall how hard fractions were 
when we first studied them, and how we avoided them in 
our calculations as much as possible. For exactly the 
same reason these ancient Chaldeans used the number 
60 as their unit wherever possible, because that number 
being divisible by more numbers than any other less than 
100, its use and the use of any six or a multiple of six 
avoided fractions. Thus they divided circles into 360 
degrees (6 X 60), each degree into 60 minutes, and each 
minute into 60 seconds. They divided the zodiac into 



142 



TIME AND THE CALENDAR 



spaces of 30° each, giving us the plan of twelve months 
in the year. Their divisions of the day led to our 24 
hours, each having 60 minutes, with 60 seconds each. 
They used the week of seven days, one for each of the 
heavenly bodies that were seen to move in the zodiac. 
This origin is suggested in the names of the days of the 
week. 

Days of the Week 



Modern 
English 


Celestial 
Origin 


Koman 


Modern 

French 


Ancient 
Saxon ■ 


Modern 
German 


1. Sunday 


Sun 


Bias Solis 


Dimanche 


Sunnan-daeg 


Sonntag 


2. Monday 


Moon 


Dies Lunae 


Lundi 


Monan-daeg 


Montag 


3. Tuesday 


Mars 


Dies Martis 


Mardi 


Mythical God 
Tiew or Tuesco 
Tues-daeg 


Dienstag 


4. Wednesday 


Mercury 


Dies Mercurii 


Mercredi 


Woden 
Woden' s-daeg 


(Mid-week) 
Mittwoche 


5. Thursday 


Jupiter 


Dies Jovis 


Jeudi 


Thor (thunderer) 
Thors-daeg 


Donnei-stag 


6. Friday 


Venus 


Dies Veneris 


Vendredi 


Friga 
Frigedaeg 


Freitag 


7. Saturday 


Saturn 


Dies Saturn! 


Samedi 


Saeter-daeg 


Samstag or 
Sonnabend 



Complex Calendar Conditions in Turkey. " But it is in 
Turkey that the time problem becomes really compUcated, 
very irritating to him who takes it seriously, very fimny 
to him who enjoys a joke. To begin with, there are four 
years in Turkey — a Mohammedan civil year, a Moham- 
medan reUgious year, a Greek or Eastern year, and a Euro- 
pean or Western year. Then in the year there are both 
lunar months depending on the changes of the moon, and 
months which, hke ours, are certain artificial proportions 
of the solar year. Then the varieties of language in 



COMPLEX CALENDAR CONDITIONS IN TURKEY 143 



Turkey still further complicate the calendars in custom- 
ary use. I brought away with me a page from the diary 
which stood on my friend's library table, and which is 
customarily sold in Turkish shops to serve the purpose of 
a calendar; and I got from my friend the meaning of the 
hieroglyphics, which I record here as well as I can remem- 
ber them. This page represents one day. Numbering the 
compartments in it from left to right, it reads as follows: 



1. March, 1318 (Civil Year). 

2. March, 1320 (ReKgious 

Year). 

3. Thirty-one days (Civil 

Year). 

4. Wednesday. 

5. Thirty days (Religious 

Year). 

6. 27 (March: Civil Year). 

7. (March: Religious Year.) 

8. March, Wednesday (Ar- 

menian). 

9. April, Wednesday(French) 

10. March, Wednesday(Greek) 

11. Ecclesiastical Day (French 

R. C. Church). 

12. March, Wednesday (Rus- 

sian). 

13. Month Day (Hebrew). 

14. Month Day (Old Style). 

15. Month Day (New Style). 

16. Ecclesiastical Day (Ar- 

menian). 

17. Ecclesiastical Day(Greek) 

18. Midday, 5:35, 1902; Mid- 

day, 5:21. 



ITIA^U ji^iTr- 









rvn 



iiaps-anrtfcuFfrh 



MAPTIOl- TETAP. 



MAPTT. CPt^A 



27 9 



AVRIL - MERCR 



S. Hugues- 



]p»3 2 CJ^pTT 



Ifl. »!■ AJ <j..4>ji 



MsT^uiri]; Man tt,; Iv OiroaXovfxt) 



ntu or 5. 35 iu^ 1902 IMidi 5 21 Jr 30 
Fig. 44 



" I am not quite clear in my mind now as to the meaning 
of the last section, but I think it is that noon according to 
European reckoning, is twenty-one minutes past five accord- 



144 TIME AND THE CALENDAR 

ing to Turkish reckoning. For there is in Turkey, added 
to the comphcation of year, month, and day, a further 
corapUcation as to hours. The Turks reckon, not from an 
artificial or conventional hour, but from sunrise, and their 
reckoning runs for twenty-four hours. Thus, when the 
sun rises at 6 : 30 our noon will be 5 : 30, Turkish time. The 
Turkish hours, therefore, change ever}' day. The steam- 
ers on the Bosphorus rim according to Turkish time, and 
one must first look in the time-table to see the hour, and 
then calculate from sunrise of the day what time by his 
European clock the boat will start. My friends in Turkey 
had apparently gotten used to this complicated calendar, 
with its variable years and months and the constantly 
changing hours, and took it as a matter of course." * 

Modem Jewish Calendar. The modern Jewish calendar 
employs also a lunar year, but has alternate years length- 
ened by adding extra days to make up the difference 
between such year and the solar year. Thus one year 
will have 354 days, and another 22 or 23 days more. 
Sept. 23, 1900, according to our calendar, was the begin- 
ning of their year 5661. 

Many remedies have been suggested for readjusting 
our calendar so that the same date shall always recur on 
the same day of the week. "\^Tniile it is interesting for the 
student to speculate on the problem and devise ways of 
meeting the difficulties, none can be suggested that does 
not involve so many changes from our present system that 
it will be impossible for a long, long time to overcome 
social inertia sufficiently to accomplish a reform. 

If the student becomes impatient with the complexity 
of the problem, he may recall with profit these words of 

* The Impressions of a Careless Traveller, by Lyman Abbott. — 
The Outlook, Feb. 28, 1903. 



TOPICS POR SPECIAL REPORTS 145 

John Fiske: "It is well to simplify things as much as 
possible, but this world was not so put together as to save 
us the trouble of using our wits." 

Three Christmases in One Year. " Bethlehem, the 
home of Christmases, is that happy Utopia of which every 
American child dreams — it has more than one Christmas. 
In fact, it has three big ones, and, strangely enough, the 
one falling on December 25th of our calendar is not the 
greatest of the three. It is, at least, the first. Thirteen 
days after the Latin has burned his Christmas incense in 
the sacred shrine, the Greek Church patriarch, observing 
that it is Christmas-time by his slower calendar, catches up 
the Gloria, and bows in the Grotto of the Nativity for the 
devout in Greece, the Balkan states, and all the Russias. 
After another period of twelve days the great Armenian 
Church of the East takes up the anthem of peace and 
good-will, and its patriarch visits the shrine." * 

Topics for Special Reports. The gnomon. The clep- 
sydra. Other ancient devices for reckoning time. The 
week. The Metonic cycle and the Golden Number. The 
calculation of Easter. The Roman calendar. Names of 
the months and days of the week. Calendar reforms. 
The calendar of the French Revolution. The Jewish 
calendars. The Turkish calendar. 

* Ernest I. Lewis in Woman's Home Companion, December, 1903. 



JO. MATH. GEO. — 10 



CHAPTER VIII 

SEASONS 

Vertical and Slanting Rays of the Sun. He would be 
unobservant, indeed, who did not know from first-hand 
experience that the morning and evening rays of the sun 
do not feel so warm as those of midday, and, if living out- 
side the torrid zone, that rays from the low winter sun in 
some way lack the heating power of those from the high 
summer sun. The reason for this difference may not be 
so apparent. The vertical rays are not warmer than the 
slanting ones, but the more nearly vertical the sun, the 
more heat rays are intercepted by a given surface. If 
you place a tub in the rain and tip it so that the rain falls 
in slantingly, it is obvious that less water will be caught 
than if the tub stood at right angles to the course of the 
raindrops. But before we take up in detail the effects of 
the shifting rays of the sun, let us carefully examine the 
conditions and causes of the shifting. 

Motions of the Earth. The direction and rate of the 
earth's rotation are ascertained from the direction and 
rate of the apparent rotation of the celestial sphere. The 
direction and rate of the earth's revolution are ascertained 
from the apparent revolution of the sun among the stars 
of the celestial sphere. Just as any change in the rotation 
of the earth would produce a corresponding change in the 
apparent rotation of the celestial sphere, so any change 
in the revolution of the earth would produce a correspond- 
ing change in the apparent revolution of the sun. 

Were the sun to pass among the stars at right angles to 

146 



EQUINOXES 147 

the celestial equator, passing through the celestial poles, 
'we should know that the earth went around the sun in a 
path whose plane was perpendicular to the plane of the 
equator and was in the plane of the axis. In such an 
event the sun at some time during the year would shine 
vertically on each point on the earth's surface. Seasons 
would be nearly the same in one portion of the earth 
as in another. The sun would sometimes cast a north 
shadow at any given place and sometimes a south shadow. 
Were the sun always in the celestial equator, the ecliptic 
coinciding with it, we should know that the earth traveled 
around the sun at right angles to the axis. The vertical 
ray of the sun would then always be overhead at noon on 
the equator, and no change in season would occur. Were 
the plane of the earth's orbit at an angle of 45° from the 
equator the ecUptic would extend half way between the 
poles and the equator, and the sun would at one time get 
within 45° of the North star and six months later 45° 
from the South star. The vertical ray on the earth would 
then travel from 45° south latitude to 45° north latitude, 
and the torrid zone wotdd be 90° wide. 

Obliquity of the Ecliptic. But we know that the vertical 
ray never gets farther north or south of the equator than 
about 231°, or nearer the poles than about 66^°. The 
plane of the ecUptic or of the earth's orbit is, then, inclined 
at an angle of 66^° to the axis, or at an angle of 23^° to 
the plane of the equator. This obliquity of the ecliptic 
varies slightly from year to year, as was shown on 
pp. 118, 288. 

Equinoxes. The sun crosses the celestial equator twice 
a year, March 20 or 21, and September 22 or 23,* varjdng 

* The reason why the date shifts lies in the construction of our 
calendar, which must fit a year of 365 days, 5 h. 48 m. 45.51 s. The time 



148 SEASONS 

from year to year, the exact date for any year being easily 
found by referring to any almanac. These dates are 
called equinoxes (equinox; wquus, equal; nox, night), for 
the reason that the days and nights are then twelve 
hours long everywhere on earth. March 21 is called 
the vernal (spring) equinox, and September 23 is called 
the autumnal equinox, for reasons obvious to those 
who live in the northern hemisphere (see Equinox in 
Glossary). 

Solstices. About the time when the sun reaches its most 
distant point from the celestial equator, for several days it 
seems neither to recede from it nor to approach it. The 
dates when the sun is at these two points are called the 
solstices (from sol, sun; and stare, to stand). June 21 is 
the summer solstice, and December 22 is the winter solstice; 
vice versa for the southern hemisphere. The same terms 
are also appUed to the two points in the ecliptic farthest 
from the equator; that is, the position of the sun on those 
dates. 

At the Equator. March ^1. Imagine you are at the 
equator March 21. Bear in mind the fact that the North 
star (strictly speaking, the north pole of the celestial 
sphere) is on the northern horizon, the South star on the 
southern horizon, and the celestial equator extends from 
due east, through the zenith, to due west. It is sunrise of 
the vernal equinox. The sun is seen on the eastern hori- 
zon ; the shadow it casts is due west and remains due west 
until noon, getting shorter and shorter as the sun rises 
higher. 

of the vernal equinox in 1906 was March 21, 7:46 a.m., Eastern 
standard time. In 1907 it occurred 365 days, 5 h. 48 m. 45.51 s. later, or 
at 1 : 35 p.m., March 21. In 1908, being leap year, it wiU occur 366 days, 
5 h. 48 m. 45.51 s. later, or at about 7 : 24 p.m., March 20. The same facts 
are true of the solstices ; they occur June 21-22 and December 22-23, 



AT THE EQUATOR 149 

Shadows. At noon the sun, being on the celestial 
equator, is directly overhead and casts no shadow, or the 
shadow is directly underneath. In the afternoon the 




Fig. 45. Illumination of the earth in twelve positions, "corresponding to months. 
The north pole is turned toward us. 

shadow is due east, lengthening as the sun approaches 
the due west point in the horizon. At this time the sun's 
rays extend from pole to pole. The circle of illumination, 
that great circle separating the lighted half of the earth 
from the half which is turned away from the sim, since it 



160 SEASONS 

extends at this time from pole to pole, coincides with a 
meridian circle and bisects each parallel. Half of each 
parallel being in the light and half in the dark, during 
one rotation every point will be in the light half a day and 
away from the sun the other half, and day and night are 
equal ever}nvhere on the globe. 

After March 21 the sun creeps back in its orbit, gradu- 
ally, away from the celestial equator toward the North 
star. At the equator the sun thus rises more and more 
toward the north of the due east point on the horizon, 
and at noon casts a shadow toward the south. As the 
sun gets farther from the celestial equator, the south noon 
shadow lengthens, and the sun rises and sets farther 
toward the north of east and west. 

On June 21 the sun has reached the point in the eclip- 
tic farthest from the celestial equator, about 23^° north. 
The vertical ray on the earth is at a corresponding distance 
from the equator. The sun is near the constellation 
Cancer, and the parallel marking the turning of the sun 
from his course toward the polestar is called the Tropic 
(from a Greek word meaning turning) of Cancer. Our 
terrestrial parallel marking the southward turning of the 
vertical ray is also called the Tropic of Cancer. At this 
date the circle of illumination extends 23|° beyond the 
north pole, and all of the parallels north of 66^° from the 
equator are entirely within this circle of illumination and 
have dayhght during the entire rotation of the earth. At 
this time the circle of illumination cuts unequally parallels 
north of the equator so that more than half of them are in 
the lighted portion, and hence days are longer than nights 
in the northern hemisphere. South of the equator the 
conditions are reversed. The circle of illumination does 
not extend so far south as the south pole, but falls short 



AT THE EQUATOR 151 

of it 23^°, and consequently all parallels south of 66^° are 
entirely in the dark portion of the earth, and it is con- 
tinual night. Other circles south of the equator are so 
intersected by the circle of illumination that less than 
half of them are in the lighted side of the earth, and the 
days are shorter than the nights. It is midwinter there. 

After June 21 gradually the sun creeps along in its 
orbit away from this northern -point in the celestial sphere 
toward the celestial equator. The circle of illumination 
again draws toward the poles, the days are more nearly 
of the same length as the nights, the noon sun is more 
nearly overhead at the equator again, until by September 
23, the autumnal equinox, the sun is again on the celestial 
equator, and conditions are exactly as they were at the 
March equinox. 

After September 23 the sun, passing toward the South 
star from the celestial equator, rises to the south of a due 
east line on the equator, and at noon is to the south of the 
zenith, casting a north shadow. The circle of illumina- 
tion withdraws from the north pole, leaving it in darkness, 
and extends beyond the south pole, spreading there the 
glad sxmshine. Days grow shorter north of the equator, less 
than half of their parallels being in the lighted half, and 
south of the equator the days lengthen and summer comes. 

On December 22 the sun has reached the most distant 
point in the ecUptic from the celestial equator toward the 
South star, 23^° from the celestial equator and 66^° from 
the South star, the vertical ray on the earth being at corre- 
sponding distances from the equator and the south pole. 
The Sim is now near the constellation Capricorn, and every- 
where within the tropics the shadow is toward the north; 
on the tropic of Capricorn the sun is overhead at noon, 
and south of it the shadow is toward the south. Here 



152 SEASONS 

the vertical ray turns toward the equator again as the sun 
creeps in the ecliptic toward the celestial equator. 

Just as the tropics are the parallels which mark the 
farthest Umit of the vertical ray from the equator, the 
polar circles are the parallels marking the farthest extent 
of the circle of illumination beyond the poles, and are the 
same distance from the poles that the tropics are from the 
equator. 

The "Width of the Zones is thus determined by the dis- 
tance the vertical ray travels on the earth, and with the 
moving of the vertical ray, the shifting of the day circle. 
This distance is in turn determined by the angle which 
the earth's orbit forms with the plane of the equator. The 
planes of the equator and ihc orbit forming an angle of 
23^°, the vertical ray travels that many degrees each side 
of the equator, and the torrid zone is 47° wide. The circle 
of illumination never extends more than 23J° beyond each 
pole, and the frigid zones are thus 23|° wide. The remain- 
ing or temperate zones between the torrid and the frigid 
zones must each be 43° wide. 

At the North Pole. Imagine you are at the north pole. 
Bear in mind the fact that the North star is always almost 
exactly overhead and the celestial equator always on the 
horizon. On March 21 the sun is on the celestial equa- 
tor and hence on the horizon.* The sun now swings 
around the horizon once each rotation of the earth, casting 
long shadows in every direction, though, being at the 
north pole, they are always toward the south.f After the 

* Speaking exactly, the sun is seen there before the spring equinox 
and after the autumnal equinox, owing to refraction and the dip of the 
horizon. See p. 160. 

t The student should bear in mind the fact that directions on the 
earth are determined solely by reference to the true geographical 
pole, not the magnetic pole of the mariner's compass. At the north 



AT THE NORTH POLE 153 

spring equinox, the sun gradually rises higher and higher 
in a gently rising spiral until at the summer solstice, June 
21, it is 23-^° above the horizon. After this date it gradu- 
ally approaches the horizon again until, September 23, 
the autumnal equinox, it is exactly on the horizon, and 
after this date is seen no more for six months. Now the 
stars come out and may be seen perpetually tracing their 
circular courses around the polestar. Because of the reflec- 
tion and refraction of ihe rays of Hght in the air, twihght 
prevails when the sun is not more than about 18° below 
the horizon, so that for only a small portion of the six 
months' winter is it dark, and even then the long journeys 
of the moon above the celestial equator, the bright stars 
that never set, and the auroras, prevent to tair' darkness 
(see p. 164). On December 22 the sun is 23^° below the 
horizon, after which it gradually approaches the horizon 
again, twihght soon setting in until March 21 again shows 
the welcome face of the sun. 

At the South Pole the conditions are exactly reversed. 
There the sun swings around the horizon in the opposite 
direction; that is, in the direction opposite the hands of a 
watch when looked at from above. The other half of the 
celestial sphere from that seen at the north pole is always 
above one, and no stars seen at one pole are visible at the 
other pole, excepting the few in a very narrow belt around 
the celestial equator, hfted by refraction of hght. 
pole the compass points due south, and at points between the magnetic 
pole and the geographical pole it may point in any direction excepting 
toward the north. Thus Admiral A. H. Markham says, in the Youth's 
Companion for June 22, 1902: 

" When, in 1876, 1 was sledging over the frozen sea in my endeavor 
to reach the north pole, and therefore traveling in a due north direc- 
tion, I was actually steering by compass E. S. E., the- variation of the 
compass in that locality varying from ninety-eight degrees to one 
hundred and two degrees westerly." 



154 SEASONS 

Parallelism of the Earth's Axis. Another condition of 
the earth in its revolution should be borne in mind in 
explaining change of seasons. The earth might rotate on 
an axis and revolve around the sun with the axis inchned 
23^° and still give us no change in seasons. This can 
easily be demonstrated by carrying a globe around a cen- 
tral object representing the sun, and by rotating the axis 
one can maintain the same inchnation but keep the verti- 
cal ray continually at the equator or at any other circle 
within the tropics. In order to get the shifting of the 
vertical ray and change of seasons which now obtain, the 
axis must constantly point in the same direction, and its 
position at one time be parallel to its position at any other 
time. This is called the parallelism of the earth's axis. 

That the earth's axis has a very slow rotary motion, a 
sUght periodic " nodding " which varies its inclination 
toward the plane of the ecliptic, and also irregular motions 
of diverse character, need not confuse us here, as they are 
either so minute as to require very delicate observations 
to determine them, or so slow as to require many years to 
show a change. These three motions of the axis are dis- 
cussed in the Appendix under " Precession of the Equi- 
noxes," " Nutation of the Poles," and " Wandering of the 
Poles " (p. 286). 

Experiments with the Gyroscope. The gyroscope, prob- 
ably familiar to most persons, admirably illustrates the 
causes of the paralleUsm of the earth's axis. A disk, sup- 
ported in a ring, is rapidly whirled, and the rotation tends 
to keep the axis of the disk always pointing in the same 
direction. If the ring be held in the hands and carried 
about, the disk rapidly rotating, it will be discovered that 
any attempt to change the direction of the axis will meet 
with resistance. This is shown in the simple fact that a 



DAY'S LENGTH AT THE EQUINOXES 155 

rapidly rotating- top remains upright and is not easily 
tipped over; and, similarly, a bicycle running at a rapid 
rate remains erect, the rapid motion of the wheel (of 
top) giving the axis a tendency to remain in the same 
plane. 

The gyroscope shown in Figure 46 * is one used by 

Professor R. S. Holway of the University of California. 

; It was made by mounting a six-inch sewing-machine wheel 

on ball bearings in the fork of an old bicycle. Its advan- 



Fig. 46 

tages over those commonly used are its simplicity, the 
ball bearings, and its greater weight. 

Foucault Experiment. In 1852, the year after his 
famous pendulimi experiment, demonstrating the rotation 
of the earth, M. Leon Foucault demonstrated the same 
facts by means of a gyroscope so mounted that, although 
j the earth turned, the axis of the rotating wheel remained 
constantly in the same direction. 

Comparative Length of Day and Night 

Day's Length at the Equinoxes. One half of the earth 
being always in the sunlight, the circle of illumination is a 
great circle. The vertical ray marks the center of the 
lighted half of the surface of the earth. At the equinoxes 

* Taken, 'by permission, from the Journal of Geography for 
February, 1904. 



156 SEASONS 

the vertical ray is at the equator, and the circle of illumi- 
nation extends from pole to pole, bisecting every parallel. 
Since at this time any given parallel is cut into two equal 
parts by the circle of illumination, one half of it is in the 
sunhght, and one half of it is in darkness, and during one 
rotation a point on a parallel will have had twelve hours 
day and twelve hours night. (No allowance is made for 
refraction or twihght.) 

Day's Length after the Equinoxes. After the vernal 
equinox the vertical ray moves northward, and the circle 
of illumination extends beyond the north pole but falls 
short of the south pole." Then all parallels, save the 
equator, are unequally divided by the circle of illumination, 
for more than half of each parallel north of the equator 
is in the light, and more than half of each parallel south 
of the equator is in darkness. Consequently, while the 
vertical ray is north of the equator, or from March 21 to 
September 23, the days are longer than the nights north 
of the equator, but are shorter than the nights south of 
the equator. 

During the other half of the year, when the vertical ray 
is south of the equator, these conditions are exactly 
reversed. The farther the vertical ray is from the equator, 
the farther is the circle of illumination extended beyond 
one pole and away from the other pole, and the more 
unevenly are the parallels divided by it; hence the days 
are proportionally longer in the hemisphere where the 
vertical ray is, and the nights longer in the opposite hemi- 
sphere. The farther from the equator, too, the greater 
is the difference, as may be observed from Figure 50, 
page 162. Parallels near the equator are always nearly 
bisected by the circle of illumination, and hence day 
nearly equals night there the year around. 



LONGEST DAYS AT DIFFERENT LATITUDES 157 

Day's Length at the Equator. How does the length of 
day at the equator compare with the length of night? 
When days are shorter south of the equator, if they are 
longer north of it and vice versa, at the equator they must 
be of the same length. The equator is always bisected 
by the circle of illumination, consequently half of it is 
always in the sunhght. This proposition, simple though 
it is, often needs further demonstration to be seen clearly. 
It will be obvious if one sees: 

(a) A point on a sphere 180° in any direction from a 
point in a great circle lies in the same circle. 

(6) Two great circles on the same sphere must cross 
each other at least once. 

(c) A point 180° from this point of intersection, common 
to both great circles, will lie in each of them, and hence 
must be a point common to both and a point of inter- 
section. Hence two great circles, extending in any 
direction, intersect each other a second time 180° from 
the first point of crossing, or half way around. The circle 
of illumination and equator are both great circles and 
hence bisect each other. If the equator is always bisected 
by the circle of illumination, half of it must always be in 
the Ught and half in the dark. 

Day's Length at the Poles. The length of day at the 
north pole is a Httle more than six months, since it extends 
from March 21 until September 23, or 186 days. At the 
north pole night extends from September 23 until March 
21, and is thus 179 days in length. It is just opposite at 
the south pole, 179 days of sunshine and 186 days of 
twilight and darkness. This is only roughly stated in full 
days, and makes no allowance for refraction of Kght or 
twilight. 

Longest Days at Different Latitudes. The length of the 



158 



SEASONS 



longest day, that is, from sunrise to sunset, in different 
latitudes is as follows: 



Lat. 


Day 


Lat. 


Day 


Lat. 


Day 


Lat. 


Day 


0° 


12 h. 


25° 


13 h. 34 m. 


50° 


16 h. 


9 m. 


70° 


65 days 


5° 


12 h. 17 m. 


30° 


13 h. 56 m. 


55° 


17 h. 


7 m. 


75" 


103 " 


10° 


12 h. 35 m. 


35° 


14 h. 22 m. 


60° 


18 h. 


30 m. 


80" 


134 " 


15° 


12 h. 53 m. 


40° 


14 h. 51m. 


65° 


21 h. 


09 m. 


85" 


161 " 


20° 


13 h. 13 m. 


45° 


15 h. 26 m. 


66° 33' 


24 h. 


00 m. 


90° 


6 mos. 



The foregoing table makes no allowance for the fact 
that the vertical ray is north of the equator for a longer 
time than it is south of the equator, owing to the fact that 
we are farther from the sun then, and consequently the 
earth revolves more slowly in its orbit. No allowance is 
made for refraction, which lifts up the rays of the sun 
when it is near the horizon, thus lengthening days every- 
where. 

Repbaction of Light 
The rays of light on entering the atmosphere are bent 




Fig. 47 



out of straight courses. Whenever a ray of light enters 
obliquely a medium of greater or of less density, the ray 
is bent out of its course (Fig. 47). Such a change in 



AMOUNT OF REFRACTION VARIES 159 

direction is called refraction. When a ray of light enters 
obliquely a medium of greater density, as in passing 
through from the upper rarer atmosphere to the lower 
denser layers, or from air into water, the rays are bent in 
the direction toward a perpendicular to the surface or less 
obUquely. This is called the first law of refraction. The 
second law of refraction is the converse of this ; that is, on 
entering a rarer medium the ray is bent more obliquely 
or away from a perpendicular to the surface. When a 




Fig. 48 

ray of light from an object strikes the eye, we see the 
object in the direction taken by the ray as it enters the 
eye, and if the ray is refracted this will not be the real 
position of the object. Thus a fish in the water (Fig. 48) 
would see the adjacent boy as though the boy were nearly 
above it, for the ray from the boy to the fish is bent 
downwards, and the ray as it enters the eye of the fish 
seems to be coming from a place higher up. 

Amount of Refraction Varies. The amount of refrac- 
tion depends upon the difference in the density of the 



160 



SEASONS 



media and the obliqueness with which the rays enter. 
Rays entering perpendicularly are not refracted at all. 
The atmosphere differs very greatly in density at different 
altitudes owing to its weight and elasticity. About one 
half of it is compressed within three miles of the surface 
of the earth, and at a height of ten miles it is so rare that 
sound can scarcely be transmitted through it. A ray of 
light entering the atmosphere obliquely is thus obhged to 
traverse layers of air of increasing density, and is refracted 
more and more as it approaches the earth. 

Effect of Refraction on Celestial Altitudes. Thus, refrac- 
tion increases the apparent altitudes of all celestial objects 

excepting those at the 
zenith (Fig. 49). The 
amount of refraction at 
the horizon is ordinarily 
36' 29"; that is to say, 
a star seen on the hori- 
zon is in reality over 
one half a degree below 
the horizon. The ac- 
tual amount of refrac- 
tion varies with the temperature, humidity, and pressure 
of the air, all of which affect its density and which must 
be taken into consideration in accurate calculations. 
Since the width of the sun as seen from the earth is about 
32', when the sun is seen just above the horizon it actually 
is just below it, and since the sun passes one degree in 
about four minutes, the day is thus lengthened about four 
minutes in the latitudes of the United States and more in 
higher latitudes. This accounts for the statement in alma- 
nacs as to the exact length of the day at the equinoxes. 
Theoretically the day is twelve hours long then, but prac- 




Fig. 49 



EEFRACTION ON CELESTIAL ALTITUDES 



161 



tically it is a few minutes longer. Occasionally there is 
an eclipse of the moon observed just before the sun has 
gone down. The earth is exactly between the sun and the 
moon, but because of refraction, both sun and moon are 
seen above the horizon. 

The sun and moon often appear flattened when near 
the horizon, especially when seen through a haze. This 
apparent flattening is due to the fact that rays from the 
lower portion are more obUque than those from the upper 
portion, and hence it is apparently Ufted up more than 
the upper portion. 

Mean Refraction Table 
{For Temperature 50° Fahr., barometric pressure 30 in.) 



Apparent 


Mean Re- 


Apparent 


Mean Re- 


Apparent 


Mean Re- 


Altitude. 


fraction. 


Altitude. 


fraction. 


Altitude. 


fraction. 


0^ 


36' 29.4" 


8° 


6' 33.3" 


26° 


1' 58.9" 


1 


24 53.6 


9 


5 52.6 


30 


1 40.6 


2 


18 25.5 


10 


5 19.2 


40 


1 9.4 


3 


14 25.1 


12 


4 27.5 


50 


48.9 


4 


11 44.4 


14 


3 49.5 


60 


33.6 


5 


9 52.0 


16 


3 20.5 


70 


21.2 


6 


8 28.0 


18 


2 57.5 


80 


10.3 


7 


7 23.8 


22 


2 23.3 


90 


00.0 



Twilight 

The atmosphere has the peculiar property of reflecting 
and scattering the rays of light in every direction. Were 
not this the case, no object would be visible out of the 
direct , sunshine, shadows would be perfectly black, our 
houses, excepting where the sun shone, would be perfectly 
dark, the blue sky would disappear and we could see the 
stars in the day time just as well as at night. Because 
of this diffusion of light, darkness does not immediately 
set in after sunset, for the rays shining in the upper air 



JO. MATH. QBO. 



•11 



162 



SEASONS 



are broken up and reflected to the lower air. This, in 
brief, is the explanation of twilight. There being practi- 
cally no atmosphere on the moon there is no twilight 
there. These and other consequences resulting from the 
lack of an atmospheric envelope on the moon are described 
on pp. 263, 264. 

Length of Twilight. Twilight is considered to last while 
the sun is less than about 18° below the horizon, though 
the exact distance varies somewhat with the condition of 
the atmosphere, the latitude, and the season of the year. 

There is thus a 
twilight zone im- 
mediately beyond 
the circle of illumi- 
nation, and outside 
of this zone is the 
true night. Figure 
50 represents these 
three portions: (1) 
the hemisphere re- 
ceiving direct rays (shghtly more than a hemisphere owing 
to refraction), (2) the belt 18° from the circle of illumina- 
tion, and (3) the segment in darkness — total save for 
starlight or moonhght. The height of the atmosphere is, 
of course, greatly exaggerated. The atmosphere above 
the line AB receives direct rays of light and reflects 
and diffuses them to the lower layers of atmosphere. 

Twilight Period Varies with Season. It will be seen from 
Figure 50 that the fraction of a parallel in the twihght 
zone varies greatly with the latitude and the season. At 
the equator the sim drops down at right angles to the 

18 
horizon, hence covers the 18° twilight zone in — — of a 




Fig. 50 



TWILIGHT NEAR THE EQUATOR 163 

day or one hour and twelve minutes. This remains prac- 
tically the same the year around there. In latitudes of 
the United States, the tviilight averages one and one-half 
hours long, being greater in midsummer. At the poles, 
twilight lasts about two and one-half months. 

Twilight Long in High Latitudes. The reason why the 
twilight lasts so long in high latitudes in the summer will 
be apparent if we remember that the sun, rising north of 
east, swinging slantingly around and setting to the north 
• of west, passes through the twilight zone at the same 
oblique angle. At latitude 48° 33' the sun passes around 
so obhquely at the summer solstice that it does not sink 18° 
below the horizon at midnight, and stays within the twi- 
hght zone from sunset to sunrise. At higlier latitudes on 
that date the sun sinks even less distance below the 
horizon. For example, at St. Petersburg, latitude 59° 56' 
30", the sun is only 6° 36' 25" below the horizon at mid- 
night June 21 and it is hght enough to read without 
artificial hght. From 66° to the pole the sun stays 
entirely above the horizon throughout the entire summer 
solstice, that being the boundary of the " land of the mid- 
night sun." 

Twilight Near the Equator. " Here comes science now 
taking from us another of our cherished beliefs — the wide 
superstition that in the tropics there is almost no twiUght, 
and that the ' sun goes down hke thunder out o' China 
'crosst the bay.' Every boy's book of adventure tells of 
travelers overtaken by the sudden descent of night, and 
men of science used to bear out these tales. Young, in 
his ' General Astronomy,' points out that ' at Quito the 
twihght is said to be at best only twenty minutes.' In 
a monograph upon ' The Duration of Twihght in the 
Tropics,' S. I. Bailey points out, by carefully verified 



164 SEASONS 

observation and experiments, that the tropics have their 
fair share of twihght. He says: 'Twilight may be said 
to last until the last bit of illuminated sky disappears from 
the western horizon. In general it has been found that 
this occurs when the sun has sunk about eighteen degrees 
below the horizon. . . . Arequipa, Peru, lies within the 
tropics, and has an elevation of 8,000 feet, and the air is 
especially pure and dry, and conditions appear to be 
exceptionally favorable for an extremely short twilight. 
On Sunday, June 25, 1899, the following observations 
were made at the Harvard Astronomical Station, which is 
situated here: The sun disappeared at 5:30 p.m., local 
mean time. At 6 p.m., thirty minutes after sunset, I 
could read ordinary print with perfect ease. At 6 : 30 p.m. 
I could see the time readily by an ordinary watch. At 
6:40 P.M., seventy minutes after sunset, the illuminated 
western sky was still bright enough to cast a faint shadow 
of an opaque body on a white surface. At 6:50 p.m., 
one hour and twenty minutes after sunset, it had dis- 
appeared. On August 27, 1899, the following observa- 
tions were made at Vincocaya. The latitude of this place 
is about sixteen degrees south, and the altitude 14,360 
feet. Here it was possible to read coarse print forty- 
seven minutes after sunset, and twilight could be seen for 
an hour and twelve minutes after the sun's disappearance.' 
So the common superstition about no twilight in the 
tropics goes to join the William Tell myth." — Harper's 
Weekly, April 5, 1902. 

Twilight Near the Pole. " It may be interesting to re- 
late the exact amount of light and darkness experienced 
during a winter passed by me in the Arctic regions within 
four hundred and sixty miles of the Pole. 

" From the time of crossing the Arctic circle until we 



VERTICAL RAYS AND INSOLATION 165 

established ourselves in winter quarters on the 3d. of 
September, we rejoiced in one long, continuous day. On 
that date the sun set below the northern horizon at mid- 
night, and the dayhght hours gradually decreased tmtil 
the sun disappeared at noon below the southern horizon 
on the 13th of October. 

" From this date until the 1st of March, a period of one 
hundred and forty days, we never saw the sun; but it 
must not be supposed that because the sun was absent we 
were hving in total darkness, for such was not the case. 
During the month following the disappearance of the sun, 
and for a month prior to its return, we enjoyed for an 
hour, more or less, on either side of noon, a glorious twi- 
light; but for three months it may be said we lived in 
total darkness, although of course on fine days the stars 
shone out bright and clear, rendered all the more brilhant 
by the reflection from the snow and ice by which we were 
surrounded, while we also enjoyed the light from the moon 
in its regular lunations. 

" On the 21st of December, the shortest day in the year, 
the sun at our winter quarters was at noon twenty degrees 
below the horizon. I mention this because the twilight 
circle, or, to use its scientific name, the crepusculum, when 
dawn begins and twiUght ends, is determined when the 
sun is eighteen degrees below the horizon. 

" On our darkest day it was not possible at noon to read 
even the largest-sized type." — Admiral A. H. Markham, 
R. N., in the Youth's Companion, June 22, 1899. 

Effect of the Shifting Rays of the Sun 

Vertical Rays and Insolation. The more nearly vertical 
the rays of the sun are the greater is the amount of heat 
imparted to the earth at a given place, not because a ver- 



166 



SEASONS 



tical ray is any warmer, but because more rays fall over 
a given area. In Figure 51 we notice that more perpen- 
dicular rays extend over a given area than slanting ones. 




Fig. 51 

We observe the morning and evening rays of the sun, 
even when falling perpendicularly upon an object, say 
through a convex lens or burning glass, are not so 
warm as those at midday. The reason is apparent 
from Figure 52, the 
slanting rays tra- 
verse through more 
of the atmosphere. 

At the summer 
solstice the sun's 
rays are more nearly 
vertical over Europe 
and the United 

Fig. 53 

States than at other 

times. In addition to the greater amount of heat received 

because of the less oblique rays, the days are longer than 




MAXIMUM HEAT TOLLOWS SUMMER SOLSTICE 167 

eights and consequently more heat is received during the 
day than is radiated off at night. This increasing length 
of day time greatly modifies the climate of regions far to 
the north. Here the long summer days accumulate enough 
heat to mature grain crops and forage plants. It is inter- 
esting to note that in many northern cities of the United 
States the maximum temperatures are as great as in some 
southern cities. 

How the Atmosphere is Heated. To imderstand how 
the atmosphere gets its heat we may use as an illustration 
the pecuhar heat-receiving and heat-transmitting proper- 
ties of glass. We aU know that glass permits heat rays 
from the sun to pass readily through it, and that the dark 
rays of heat from the stove or radiator do not readily pass 
through the glass. Were it not for this fact it would be 
no warmer in a room in the sunshine than in the shade, 
and if glass permitted heat to escape from a room as 
readily as it lets the sunshine in we should have to dis- 
pense with windows in cold weather. Stating this in 
more technical language, transparent glass is diatherman- 
ous to luminous heat rays but athermanous to dark rays. 
Dry air possesses this same pecuhar property and permits 
the luminous rays from the sun to pass readily through to 
the earth, only about one fourth being absorbed as they 
pass through. About three fourths of the heat the atmos- 
phere receives is that which is radiated back as dark rays 
from the earth. Being athermanous to these rays the heat 
is retained a considerable length of time before it at length 
escapes into space. It is for this reason that high alti- 
tudes are cold, the atmosphere being heated from the 
bottom upwards. 

Maximum Heat Follows Summer Solstice. Because of 
these conditions and of the convecting currents of air, and, 



168 SEASONS 

to a very limited extent, of water, the heat is so distril> 
uted and accumulated that the hottest weather is in the 
month following the summer solstice (July in the northern 
hemisphere, and January in the southern) ; conversely, the 
coldest month is the one following the winter solstice. 
This seasonal variation is precisely parallel to the diurnal 
change. At noon the sun is highest in the sky and pours 
in heat most rapidly, but the point of maximum heat is 
not usually reached imtil the middle of the afternoon, when 
the accumulated heat in the atmosphere begins gradually 
to disappear. 

Astronomical and Climatic Seasons. Astronomically 
there are four seasons each year: spring, from the vernal 
equinox to the summer solstice; summer, from the sum- 
mer solstice to the autvunnal equinox; autumn, from the 
autumnal eqtiinox to the winter solstice; winter, from the 
winter solstice to the spring equinox. As treated in phy- 
sical geography, seasons vary greatly in number and 
length with differing conditions of topography and posi- 
tion in relation to winds, mountains, and bodies of water. 
In most parts of continental United States and Europe 
there are four fairly marked seasons: March, April, and 
May are called spring months; June, July, and August, 
summer months; September, October, and November, 
autumn months; and December, January, and February, 
winter months. In the southern states and in western 
Europe the seasons just named begin earlier. In Gah- 
fornia and in most tropical regions, there are two seasons, 
one wet and one dry. In northern South America there 
are four seasons, — two wet and two dry. 

From the point of view of mathematical geography 
there are four seasons having the following lengths in the 
northern hemisphere : 



HEMISPHERES UNEQUALLY HEATED 



169 



Spring: Vernal equinox . . . March 21 ) „„ , 

Summer solstice . . . June 21 ) ^^ "-^^^ 



. June 21 ) „ . J 

Autumnal equinox . . Sept. 23 ) ^^ J 

Autumn: Autumnal equinox . . Sept. 23 ) „„ j ^ 

Winter solstice . . . Dec. 22^°'^'*^^ 



Wintek: Winter solstice . . . Dec. 22 I „„ , 

Vernal equinox . . . March 21 J °^ "^^^ 



Summer half 
' 186 days. 



Winter half 179 
days. 



Hemispheres Unequally Heated. For the southern 
hemisphere, spring should be substituted for autumn, and 
summer for winter. From the foregoing it will be seen 
that the northern hemisphere has longer summers and 
shorter winters than the southern hemisphere. Since the 
earth is in perihelion, nearest the sun, December 31, the 
earth as a whole then receives more heat than in the north- 
ern summer when the earth is farther from the sun. 
Though the earth as a whole must receive more heat in 
December than in July, the northern hemisphere is then 
turned away from the sun and has its winter, which is thus 
warmer than it would otherwise be. The converse is true 
of the northern summer. The earth then being in apheUon 
receives less heat each day, but the northern hemisphere 
being turned toward the sun then has its summer, cooler 
than it would be were this to occur when the earth is in 
perihelion. It is well to remember, however, that while 
the earth as a whole receives more heat in the half 
year of perihelion, there are only 179 days in that 
portion, and in the cooler portion there are 186 days, 
so that the total amount of heat received in each 
portion is exactly the same. (See Kepler's Second Law, 
p. 284.) 



170 



SEASONS 



Determination of Latitude from Sun's Meridian 
Altitude 

In Chapter II we learned how latitude is determined by 
ascertaining the altitude of the celestial pole. We are now 
in a position to see how this is commonly determined by 
reference to the noon sun. 

Relative Positions of Celestial Equator and Celestial Pole. 
The meridian altitude of the celestial equator at a given 
place and the altitude of the celestial pole at that place are 
complementary angles, that is, together they equ&,l 90°. 
Though when understood this proposition is exceedingly 
simple, students sometimes only partially comprehend it, 
and the later conclusions are consequently hazy. 

1. The celestial equator is always 90° from the celestial 
pole. 

2. An are of the celestial sphere from the northern hori- 



^0" 


, zenith 








/( 


\ 


X -o 


A 


\ 


\cio^ 


A 






X vV * 


/ 
1 


K 


y[ 


w 


1 




•flT 


\ 


i 


k 


°M 





Hor'iTon Line at Latitude ^O^N 



Fig. 53 



zon through the zenith to the southern horizon comprises 
180°. 

3. Since there are 90° from the pole to the equator, 
from the northern horizon to the pole and from the 
southern horizon to the equator must together equal 90°. 



DECLINATION OF THE SUN 171 

One of the following statements is incorrect. Find 
which one it is. 

a. In latitude 30° the altitude of the celestial pole is 
30° and that of the celestial equator is 60°. 

h. In latitude 36° the altitude of the celestial equator 
is 54°. 

c. In latitude 48° 20' the altitude of the celestial equa- 
tor is 41° 40'. 

d. If the celestial equator is 51° above the southern 
horizon, the celestial pole is 39° above the northern horizon. 

e. If the altitude of the celestial equator is 49° 31', the 
latitude must be 40'' 29'. 

/. If the altitude of the celestial equator is 21° 24',' the 
latitude is 69° 36'. 

On March 21 the stin is on the celestial equator.* If on 
this day the sun's noon shadow indicates an altitude of 
40° we know that is the altitude of the celestial equator, 
and this subtracted from 90° equals 50°, the latitude of 
the place. On September 23 the sun is again on the celes- 
^ tial equator, and its noon altitude subtracted from 90° 
equals the latitude of the place where the observation is 
made. 

Declination of the Sun. The declination of the sun or of 
any other heavenly body is its distance north or south of 
the celestial eqimtor. The analemma, shown on page 127, 
gives the approximate dechnation of the sun for every day 
in the year. The Nautical Almanac, Table 1, for any 

* Of course; the center of the sun is not on the celestial equator all 
day, it is there but the moment of its crossing. The vernal equinox 
is the point of crossing, but we commonly apply the term to the day 
when the passage of the sun's center across the celestial equator 
occurs. During this day the sun travels northward less than 24', 
and since its diameter is somewhat more than 33' some portion of the 
sun's disk is on the celestial equator the entire day. 



172 



SEASONS 



month gives the decUnation very exactly (to the tenth of 
a second) at apparent sun noon at the meridian of Green- 
wich, and the difference in declination for every hour, so 
the student can get the dechnation at his own longitude 
for any given day very exactly from this table. With- 
out good instruments, however, the proportion of error of 
observation is so great that the analemma will answer 
ordinary purposes. 

How to Determine the Latitude of Any Place. By ascer- 
taining the noon altitude of the sun, and referring to the 

analemma or a declination 
table, one can easily compute 
the latitude of a place. 

1. First determine when 
the sun will be on your me- 
ridian and its shadow strike 
a north-south line. This is 
discussed on pp. 128, 129. 

2. By some device meas- 
ure the altitude of the sun at 
apparent noon; i.e., when the 
shadow is north. A card- 
board placed level imder a 
window shade, as illustrated 
in Figure 54, will give sur- 
prisingly accurate results; a 
carefuUy mounted quadrant 
(see Fig. 55), however, will 

give more uniformly successful measurements. Angle A 
(Fig. 54), the shadow on, the quadrant, is the altitude of 
the Sim. This is apparent from Figure 56, since xy is the 
hne to the sun, and angle B = angle A. 

3. Consult the analenima and ascertain the declination 




Fig. 54 



EXAMPLE 



173 



of the sun. Add this to the sun's altitude if south dechna- 
tion, and subtract it if north declination. If you are south 
of the equator you 
must subtract dec- 
lination south and 
add declination 
north. (If the ad- 
dition makes the 
altitude of the sun 
more than 90° sub- 
tract 90° from it, 
as imder such cir- 
cumstances you are 
north of the equa- 
tor if it is a south 
shadow, or south of 
the equator if it 
is a north shadow. 
This will occur only 
within the tropics.) 

4. Subtract the result of step three from 90°, and the 

remainder is your latitude. 
Example. For example, 
say you are at San Fran- 
cisco, October 23, and wish 
to ascertain your latitude. 

1. Assume you have a 
north-south line. (The sun's 
shadow will cross it on that 
date at 11 h. 54 m. 33 s., 
A.M., Pacific time.) 

2. The altitude of the 
sun when the shadow is north is found to be 41°. 




Fig. 55 




Hg. 56 



174 



SEASONS 



3. The declination is 11° S. Adding we get 52°, the 
altitude of the celestial equator. 

4. 90° — 52° equals 38°, latitude of place of observer. 
Conversely, knowing the latitude of a place, one can 

ascertain the noon altitude of the sun at any given day. 
From the analemma and the table of latitudes many inter- 
esting problems will suggest themselves, as the following 
examples illustrate. 

Problem. 1. How high above the horizon does the sun 
get at St. Petersburg on December 22? 

Solution. The latitude of St. Petersburg is 59° 56' N., 

hence the altitude of the celestial equator is 30° 4'. The 

declination of the sun December 22 is 23° 27' S. Since 

south is below the celestial equator 

at St. Petersburg, the altitude of the 

sun is 30° 4' less 23° 27', or 6° 37'. 

Problem. 2. At which city is the 

noon sun higher on June 21, Chicago 

or Quito? 

Solution. The latitude of Chicago 
is 41° 50', and the altitude of the 
celestial equator, 48° 10'. The dec- 
Unation of the sun June 21 is 23° 
27' N. North being higher than the 
celestial equator at Chicago, the 
noon altitude of the sun is 48° 10' 
plus 23° 27', or 71° 37'. 

The latitude of Quito being 0°, the 
altitude of the celestial equator is 
90°. The declination of the sun being 
23° 27' from this, the sun's noon alti- 
tude must be 90° less 23° 27', or 66° 33'. The sun is thus 
5° 4' higher at Chicago than at the equator on June 21. " 




Fig. 57- Taking the altitude 
of the sun at sea. 



LATITUDE FROM MOON OR STARS 175 

Latitude from Moon or Stars. With a more extended 
knowledge of astronomy and mathematics and with suitable 
instrmnents, we might ascertain the position of the celes- 
tial equator in the morning or evening from the moon, 
planets, or stars as well as from the sun. At sea the 
latitude is commonly ascertained by making measurements 
of the altitudes of the sun at apparent noon with the sex- 
ta,nt. The declination tables are used, and allowances 
are made for refraction and for the " dip " of the horizon, 
and the resultant calculation usually gives the latitude 
within about half a mile. At observatories, where the 
latitude must be ascertained with the minutest precision 
possible, it is usually ascertained from star observations 
with a zenith telescope or a '"' meridian circle " telescope, 
and is verified in many ways. 



CHAPTER IX 

TIDES 

Tides and the Moon. The regular rise and fall of the 
level of the sea and the accompanying inflows and out- 
flows of streams, bays, and channels, are called tides. Since 
very ancient times this action of the water has been asso- 
ciated with the moon because of the regular interval 
elapsing between a tide and the passage of the moon over 
the meridian of the place, and a somewhat uniform increase 
in the height of the tide when the moon in its orbit around 
the earth is nearest the sun or is farthest from it. This 
unquestioned lunar influence on the ocean has doubtless 
been responsible as the basis for thousands of unwarranted 
associations of cause and effect of weather, vegetable 
growth, and even human temperament and disease with 
phases of the moon or planetary or astral conditions. 

Other Periodic Ebbs and Flows. Since there are other 
periodical ebbs and flows due to various causes, it may be 
well to remember that the term tide properly appKes only 
to the periodic rise and fall of water due to unbalanced 
forces in the attraction of the siui and moon. Other con- 
ditions which give rise to more or less periodical ebbs and 
flows of the oceans, seas, and great lakes are : 

a. Variation in atmospheric pressure; low barometer 
gives an upUft to water and high barometer a depression. 

h. Variabihty in evaporation, rainfall and melting snows 
produces changes in level of adjacent estuaries. 

c. Variability in wind direction, especially strong and 

continuous seasonal winds like monsoons, lowers the 

176 



MOON'S ORBIT 177 

level on the leeward of coasts and piles it up on the wind- 
ward side. 

d. Earthquakes sometimes cause huge waves. 

A few preliminary facts to bear in mind when consider- 
ing the causes of tides: 

The Moon 

Sidereal Month. The moon revolves around the earth 
in the same direction that the earth revolves about the 
sun, from west to east. If the moon is observed near a 
given star on one night, twenty-four hours later it will be 
found, on the average, about 13.2° to the eastward. To 
reach the same star a second time it will require as many 
days as that distance is contained times in 360° or about 
27.3 days. This is the sidereal month, the time required 
for one complete revolution of the moon. 

Sjoiodic Month. Suppose the moon is near the sun at a 
given time, that is, in the same part of the celestial sphere. 
During the twenty-fours hours following, the moon will 
creep eastward 13.2° and the sun 1°. The moon thus 
gains on the sun each day about 12.2°, and to get in con- 
junction with it a second time it will take as many days 
as 12.2° is contained in 360° or about 29.5 days. This is 
called a synodic (from a Greek word meaning " meeting ") 
month, the time from conjunction with the sun — new 
moon — until the next conjunction or new moon. The 
term is also applied to the time from opposition or full 
moon until the next opposition or full moon. If the 
phases of the moon are not clearly understood it would 
be well to foUow the suggestions on this subject in the 
first chapter. 

Moon's Orbit. The moon's orbit is an ellipse, its 



178 TIDES 

nearest point to the earth is called perigee (from peri, 
around or near; and ge, the earth) and is about 221,617 
miles. Its most distant point is called apogee (from apo, 
from; and ge, earth) and is about 252,972 miles. The 
average distance of the moon from the earth is 238,840 
miles. The moon's orbit is incHned to the ecliptic 5° 8' 
and thus may be that distance farther north or south than 
the sun ever gets. 

The new moon is said to be in conjunction with the sun, 
both being on the same side of the earth. If both are 
then in the plane of the ecliptic an ecUpse of the sun must 
take place. The moon being so small, relatively (diam- 
eter 2,163 miles), its shadow on the earth is small and 
thus the eclipse is visible along a relatively narrow path. 

The full moon is said to be in opposition to the sun, 
it being on the opposite side of the earth. If, when in 
opposition, the moon is in the plane of the ecliptic it will 
be echpsed by the shadow of the earth. AVhen the moon 
is in conjunction or in opposition it is said to be in syzygy. 

Gravitation 

Laws Restated. This force was discussed in the first 
chapter where the two laws of gravitation were explained 
and illustrated. The term gravity is appHed to the force 
of gravitation exerted by the earth (see Appendix, p. 279). 
Since the explanation of tides is simply the apphcation of 
the laws of gravitation to the earth, sun, and moon, we 
may repeat the two laws ; 

First law: The force of gravitation varies directly as 
the mass of the object. 

Second law : The force of gravitation varies inversely 
as the square of the distance of the object. 



SUN'S ATTRACTION 179 

Sun's Attraction Greater, but Moon's Tide-Producing 
Influence Greater. There is a widely cxirrent notion that 
since the moon causes greater tides than the sun, in the 
ratio of 5 to 2, the moon must have greater attractive 
influence for the earth than the sun has. Now this cannot 
be true, else the earth would swing around the moon as 
certainly as it does arotmd the sun. Applying the laws of 
gravitation to the problem, we see that the sun's attrac- 
tion for the earth is approximately 176 times that of the 
moon.* 

The reasoning which often leads to the erroneous con- 
clusion just referred to, is probably something like this: 

Major premise: Lunar and solar attraction causes tides. 

Minor premise: Lunar tides are higher than solar tides. 

' Conclusion : Lunar attraction is greater than solar 
attraction. 

We have just seen that the conclusion is in error. One 
or both of the premises must be in error also. A study 
of the causes of tides will set this matter right. 

Causes of Tides f 

It is sometimes erroneously stated that wind is caused 
by heat. It would be more nearly correct to say that 
wind is caused by the unequal heating of the atmosphere. 
Similarly, it is not the attraction of the sun and moon 
for the earth that causes tides, it is the unequal attraction 
for different portions of the earth that gives rise to un- 
balanced forces which produce tides. 

* For the method of demonstration, see p. 19. The following data 
are necessary: Earth's mass, I; sun's mass, 330,000; moon's mass, ^; 
distance of earth to sun, 93,000,000 miles; distance of earth to moon, 
239,000 miles. 

t A mathematical treatment will be found in the Appendix. 



180 



TIDES 



Portions of the earth toward the moon or sun are 8,000 
miles nearer than portions on the side of the earth opposite 
the attracting body, hence the force of gravitation is 
shghtly different at those points as compared with other 
points on the earth's surface. It is obvious, then, that at 
A and B (Fig. 58) there are two unbalanced forces, that 
is, forces not having counterparts elsewhere to balance 
them. At these two sides, then, tides are produced. 




To Sun or Moon 
>»? > 



Fig. 58 



since the water of the oceans yields to the influence of 
these forces. That this may be made clear, let us examine 
these tides separately. 

The Tide on the Side of the Earth Toward the Moon. If A 
is 239,000 miles from the moon, B is 247,000 miles away 
from it, the diameter of the earth being AB (Fig. 58). 
Now the attraction of the moon at A and B is away 
from the center of the earth and thus lessens the force of 
gravity at those points, lessening more at A since A is 
nearer and the moon's attraction is exerted in a line 



SOLAR TIDES COMPARED WITH LUNAR TIDES 181 

directly opposite to that of gravity. The water, being 
fluid and easily moved, yields to this lightening of its 
-weight and tends to " pile up under the moon." We thus 
have a tide on the side of the earth toward the moon. 

Tidal Wave Sweeps Westward. As the earth turns 
on its axis it brings successive portions of the earth toward 
the moon and this wave sweeps around the globe as nearly 
as possible under the moon. The tide is retarded some- 
what by shallow water and the configuration of the coast 
and is not found at a given place when the moon is at 
meridian height but lags somewhat behind. The time 
between the passage of the moon and high tide is called 
the establishment of the port. This time varies greatly at 
different places and varies somewhat at different times of 
the year for the same place. 

Solar Tides Compared with Lunar Tides. Solar tides 
are produced on the side of the earth toward the sun for 
exactly the same reason, but because the sun is so far 
away its attraction is more uniform upon different parts 
of the earth. If A is 93,000,000 miles from the sun, B is 
93,008,000 miles from the sun. The ratio of the squares 
of these two numbers is much nearer unity than the ratio 
of the numbers representing the squares of the distances 
of A and B from the moon. If the sun were as near as 
the moon, the attraction for A would be greater by ari 
enormous amount as compared with its attraction for B. 
Imagine a ball made of dough with lines connected to 
every particle. If we pull these lines uniformly the ball 
will not be pulled out of shape, however hard we pull. If, 
however, we pull some lines harder than others, although 
we pull gently, will not the ball be pulled out of shape? 
Now the pull of the sun, while greater than that of the 
moon, is exerted quite evenly throughout the earth and 



182 TIDES 

has but a slight tide-producing power. The attraction of 
the moon, while less than that of the sun, is exerted less 
evenly than that of the sun and hence produces greater 
tides. 

It has been demonstrated that the tide-producing force 
of a body varies inversely as the cube of its distance and 
directly as its mass. Applying this to the moon and sun 
we get: 

Let T = sun's tide-producing power, 
and t = moon's tide-producing power. 

The sun's mass is 26,500,000 times the moon's mass, 

:.T -A:: 26,500,000 : 1. 

But the sun's distance from the earth is 390 times the 
moon's distance, 

■■1 .t.. gg^3 . 1. 

Combining the two proportions, we get, 
T :t ■.:2 -.5. 

It has been shown that, owing to the very nearly equal 
attraction of the sun for different parts of the earth, a 
body's weight is decreased when the sun is overhead, as 
compared with the weight six hours from then, by only 

; that is, an object weighing a ton varies in 

.ZUiUUU.UUU 

weight I of a grain from sunrise to noon. In case of the 
moon this difference is about 2^ times as great, or nearly 
2 grains. 



THE TIDE ON SIDE OF EARTH OPPOSITE MOON 183 

Tides on the Moon. It may be of interest to note that 
the effect of the earth's attraction on different sides of the 
moon must be twenty times as great as tliis, so it is 
thought that when the moon was warmer and had oceans * 
the tremendous tidal waves swinging around in the oppo- 
site direction to its rotation caused a gradual retardation 
of its rotation until, as ages passed, it came to keep the 
same face toward the earth. The planets nearest the sun, 
Mercury and Venus, probably keep the same side toward 
the Sim for a similar reason. Appljdng the same reasoning 
to the earth, it is believed that the period of rotation must 
be gradually shortening, though the rate seems to be 
entirely inappreciable. 

The Tide on the Side of the Earth Opposite the Moon. A 
planet revolving around the sun, or a moon about a planet, 
takes a rate v/hich varies in a definite mathematical ratio 
to its distance (see p. 285). The sun ptdls the earth to- 
ward itself about one ninth of an inch every second. If 
the earth were nearer, its revolutionary motion would be 
faster. In case of planets having several satellites it is 
observed that the nearer ones revolve about the planet 
faster than the outer ones (see p. 255). Now if the 
earth were divided into three separate portions, as in 
Figure 59, the ocean nearest the sun, the earth proper, 
and the ocean opposite the sun would have three separate 
motions somewhat as the dotted lines show. Ocean A 
would revolve faster than earth C or ocean B. If these 
three portions were connected by weak bands their stretch- 
ing apart would cause them to separate entirely. The 

* The presence of oceans or an atmosphere is not essential to the 
theory, indeed, is not usually taken into account. It seems most cer- 
tain that the earth is not perfectly rigid, and the theoiy assumes that 
the planets and the moon have suflBcient viscosity to produce body 
tides. 



184 



TIDES 



tide-producing power at B is this tendency it has to fall 
away, or 'more strictly speaking, to fall toward the sun less 
rapidly than the rest of the earth. 

Moon and Earth Revolve About a Common Center of 
Gravity. What has been said of the earth's annual rev- 
olution around the sun applies equally to the earth's 




monthly swing around the center of gravity common to 
the earth and the moon. We commonly say the earth 
revolves about the sun and the moon revolves about the 
earth. Now the earth attracts the sun, in its measure, 
just as tnily as the sun attracts the earth; and the moon 
attracts the earth, in the ratio of its mass, as the earth 
attracts the moon. Strictly speaking, the earth and sun 
revolve around their common center of gravity and the 
moon and earth revolve around their center of gravity. 



COURSE OF THE TIDAL WAVE 185 

It is as if the earth were connected with the moon by 
a rigid bar of steel (that had no weight) and the two, 
thus firmly bound at the ends of this rod 239,000 miles 
long, were set spinning. If both were of the same weight, 
they would revolve about a point equidistant from each. 
The weight of the moon being somewhat less than -^^ that 
of the earth, this center of gravity, or point of balance, 
is only about 1,000 miles from the earth's center. 

Spring Tides. "When the sun and moon are in con- 
junction, both on the same side of the earth, the unequal 
attraction of both for the side toward them produces an 
unusually high tide there, and the increased centrifugal 
force at the side opposite them also produces an unusual 
high tide there. Both solar tides and both lunar tides 
are also combined when the sun and moon are in opposi- 
tion. Since the sun and moon are in syzygy (opposition 
or conjunction) twice a month, high tides, called spring 
tides, occur at every new moon and at every full moon. 
If the moon should be in perigee, nearest the earth, at the 
same time it was new or full moon, spring tides would be 
unusually high. 

Neap Tides. When the moon is at first or last quarter 
— moon, earth, and sun forming a right angle — the solar 
tides occur in the trough of the lunar tides and they are 
not as low as usual, and lunar tides occurring in the trough 
of the solar tides are not so high -as usual. 

Course of the Tidal Wave. While the tidal wave is gen- 
erated at any point under or opposite the sun or moon, it 
is out in the southern Pacific Ocean that the absence of 
shallow water and land areas offers least obstruction to its 
movement. Here a general hfting of the ocean occurs, 
and as the earth rotates the lifting progresses under or 
opposite the moon or sun from east to west. Thus a huge 



186 



TIDES 



wave with crest extending north and south starts twice a 
day off the western coast of South America. The general 
position of this crest is shown on the co-tidal map, one Une 
for every hour's difference in time. The tidal wave is 
retarded along its northern extremity, and as it sweeps 
along the coast of northern South America and North 
America, the wave assumes a northwesterly direction and 
sweeps down the coast of Asia at the rate of about 850 




Fig. 60. Co-tidal lines 



miles per hour. The southern portion passes across the 
Indian Ocean, being retarded in the north so that the 
southern portion is south of Africa when the northern por- 
tion has just reached southern India. The time it has 
taken the crest to pass from South America to south Africa 
is about 30 hours. Being retarded by the African coast, 
the northern portion of the wave assumes an almost north- 
erly direction, sweeping up the Atlantic at the rate of about 
700 miles an hour. It moves so much faster northward in 
the central Atlantic than along the coasts that the crest 



TIME BETWEEN SUCCESSIVE TIDES 187 

bends rapidly northward in the center and strikes all points 
of the coast of the United States within two or three hours 
of the same time. To reach France the wave must swing 
aroimd Scotland and then southward across the North Sea, 
reaching the mouth of the Seine about 60 hours after 
starting from South America. A new wave being formed 
about every 12 hours, there are thus several of these tidal 
waves following one another across the oceans, each 
shghtly different from the others. 

While the term " wave " is correctly applied to this tidal 
movement it is very Uable to leave a wrong impression 
upon the minds of those who have never seen the sea. 
When thinking of this tidal wave sweeping across the 
ocean at the rate of several hundred miles per hour, we 
should also bear in mind its height and length (by height 
is meant the vertical distance from the trough to the crest, 
and by length the distance from crest to crest). Out in 
midocean the height is only a foot or two and the length 
is hundreds of nules. Since the wave requires about three 
hours to pass from trough to crest, it is evident that a ship 
at sea is hfted up a foot or so during six hours and then as 
slowly lowered again, a motion not easily detected. On 
the shore the height is greater and the wave-length shorter, 
for about six hours the water gradually rises and then for 
about six hours it ebbs away again. Breakers, bores, and 
unusual tide phenomena are discussed on p. 189. 

Time Between Successive Tides. The time elapsing 

from the passage of the moon across a meridian until it 

crosses the same meridian again is 24 hours 51 min.* This, 

* More precisely, 24 h. 50 m. 51 s. This is the mean lunar day, or 
interval between successive passages of the moon over a given meridian. 
The apparent lunar day varies in length from 24 h. 38 m. to 25 h. 5 m. 
for causes somewhat similar to those producing a variation in the length 
of the apparent solar day. 



188 TIDES 

in contradistinction to the solar and sidereal day, may 
be termed a lunar day. It takes the moon 27.3 solar days 
to revolve aroimd the earth, a sidereal month. In one 
day it journeys gW of a day or 51 minutes. So if the 
moon was on a given meridian at 10 a.m., on one day, 
by 10 A.M. the next day the moon would have moved 
12.2° eastward, and to direct the same meridian a second 
time toward the moon it takes on the average 51 minutes 
longer than 24 hours, the actual time varying from 38 m. to 




Fig. 61. Low tide 

1 h. 6 m. for various reasons. The tides of one day, then, 
are later than the tides of the preceding day by an average 
interval of 51 minutes. 

In studjdng the movement of the tidal wave, we observed 
that it is retarded by shallow water. The spring tides 
being higher and more powerful move faster than the 
neap tides, the interval on successive days averaging only 
38 minutes. Neap tides move slower, averaging somewhat 
over an hour later from day to day. The establishment 
of a port, as previously explained, is the average time 



BORE OF THE AMAZON 189 

elapsing between the passage of the moon and the high 
tide following it. The establishment for Boston is 11 hours, 
27 minutes, although this varies half an hour at different 
times of the year. 

Height of Tides. The height of the tide varies greatly 
in different places, being scarcely discernible out in mid- 
ocean, averaging only 1|- feet in the somewhat sheltered 
Gulf of Mexico, but averaging 37 feet in the Bay of Fundy. 
The shape and situation of some bays and mouths of 
rivers is such that as the tidal wave enters, the front part 
of the wave becomes so steep that huge- breakers form 
and roU up the bay or river with great speed. These 
bores, as they are called, occur in the Bay of Fundy, in 
the Hoogly estuary of the Ganges, in that of the Dor- 
dogne, the Severn, the Elbe, the Weser, the Yangtze, the 
Amazon, etc. 

Bore of the Amazon. A description of the bore of the 
Amazon, given by La Condamine in the eighteenth cen- 
tury, gives a good idea of this phenomenon. " Dm-ing three 
days before the new and full moons, the period of the 
highest tides, the sea, instead of occupying six hours to 
reach its flood, swells to its highest Umit in one or two 
minutes. The noise of this terrible flood is heard five or 
six miles, and increases as it approaches. Presently you 
see a hqtiid promontory, 12 or 15 feet high, followed by 
another, and another, and sometimes by a fourth. These 
watery mountains spread across the whole channel, and 
advance with a prodigious rapidity, rending and crush- 
ing everything in their way. Immense trees are instantly 
uprooted by it, and sometimes whole tracts of land are 
swept away." 



CHAPTER X 



MAP PROJECTIONS 



To represent the curved surface of the earth, or any 
portion of it, on the plane surface of a map, involves 
serious mathematical difficulties. Indeed, it is impossible 

to do so with perfect ac- 
curacy. The term pro- 
jection, as applied to the 
representation on a plane 
of points corresponding 
to points on a globe, is 
not always used in 
geography in its strictly 
mathematical sense, but 
denotes any representa- 
tion on a plane of paral- 
lels and meridians of the 
earth. 

The Orthographic 
Projection 

This is, perhaps, the 
most readily understood 
projection, and is one of 
the oldest known, having 
been used by the ancient 
Greeks for celestial representation. The globe truly repre- 
sents the relative positions of points on the earth's surface. 

190 




Fig. (a 



PARALLELS AND MERIDIANS 



191 



It might seem that a photograph of a globe would 

correctly represent 

on a flat surface ^*«^S 

the curved surface 

of the earth. A 

glance at Figure 62, 

from a photograph 

of a globe, shows 

the parallels near 

the equator to be 

farther apart than 

those near the 

poles. This is not 

the way they are 

on the globe. The 

OrthograDhic DrO- ^'^' *^' Equatorial orthographic projection 

jection is the representation of the globe as a photo- 
graph would show it from a great distance. 





Fig. 64 



Parallels and Meridians Farther Apart in Center of Map. 

Viewing a globe from a distance, jve observe that par- 



192 



MAP PROJECTIONS 



allels and meridians appear somewhat as represented in 
Figure 63, being farther apart toward the center and in- 
creasingly nearer toward the outer portion. Now it is 
obvious from Figure 64 that the farther the eye is placed 
from the globe, the less will be the distortion, although a 
removal to an infinite distance will not obviate all distor- 
tion. Thus the eye at x sees lines to E and F much 
nearer together than lines to A and B, but the eye at the 
greater distance sees less difference. 

A^Tien the rayc are perpendicular to the axis, as in Figure 
65, the parallels at A, B, C, D, and E will be seen on the 
tangent plane XY 
at A', B', C, D', 
and E'. While the 
distance from A to 
B on the globe 
is practically the 
same as the dis- 
tance from D to E, 
to the distant eye 
A' and B' will ap- 
pear much nearer 
together than D' 
and E'. Since A 
(or A') represents a pole and E (or E') the equator, line 
XY is equivalent to a central meridian and points A', B', 
etc., are where the parallels cross it. 

How to Lay off an Equatorial Orthographic Projection. 
If parallels and meridians are desired for every 15°, divide 
the circle into twenty-four equal parts; any desired number 
of parallels and meridians, of course, may be drawn. Now 
connect opposite points with straight lines for parahels 
(as in Fig. 65). The reason why parallels are straight lines 




Fig. 65 



TO LAY OFF AN EQUATORIAL PROJECTION 193 



in the equatorial orthographic projection is apparent if 
one remembers that if the eye is in the plane of the equator 
and is at an infinite distance, the parallels will he in practi- 
cally the same plane as the eye. 

To lay off the meridians, mark on the equator points 
exactly as on the central meridian where parallels intersect 
it. The meridians may now be made as arcs of circles 
passing through the poles and these points. With one 
foot of the compasses in the equator, or equator extended, 
place the other so that 
it will pass through the 
poles and one of these 
points. After a little 
trial it will be easy to 
lay off each of the meri- 
dians in this way. 

To be strictly correct 
the meridians should not 
be arcs of circles as just 
suggested but should be 
semj-elUpses with the 
central meridian as 
major axis as shown in 
Figure 66. While somewhat more difficult, the student 
should learn how thus to lay them off. To construct the 
ellipse, one must first locate the foci. This is done by taking 
half the major axis (central meridian) as radius and with 
the point on the equator through which the meridian is 
to be constructed as center, describe an arc cutting the 
center meridian on each side of the equator. These points 
of intersection on the central meridian are the foci of the 
ellipse, one half of which is a meridian. By placing a pin 
at each of the foci and also at the point in the equator 

JO. MATH. GEO. 13 




Fig. 66. Western hemisphere, in equatorial 
orthographic projection 



194 



MAP PROJECTIONS 



where the meridian must cross and tjdng a string as a loop 
around these three pins, then withdrawing the one at the 
equator, the eUipse may be made as described in the first 
chapter. 

How to Lay off a Polar Orthographic Projection. This 
is laid off more easily than the former projection. Here 
the eye is conceived to be directly above a pole and the 

equator is the boimdary 
of the hemisphere seen. 
It is apparent that 
from this position the 
equator and parallels 
will appear as circles 
and, since the planes 
of the meridians pass 
through the eye, each 
meridian must appear 
as a straight line. 

Lay off for the equa- 
tor a circle the saine 
size as the preceding 
one (Fig. 65), sub- 
dividing it into twenty-four parts, if meridians are desired 
for every 15°. Connect these points with the center, 
which represents the pole. On any diameter mark off 
distances as on the center meridian of the equatorial 
orthographic projection (Fig. 65). Through these points 
draw circles to represent parallels. You will then have 
the complete projection as in Figure 67. 

Projections may be made with any point on the globe as 
center, though the hmits of this book will not permit the 
rather difficult explanation as to how it is done for lati- 
tudes other than 0° or 90°. With the parallels and 




Fig. 67. Polar orthographic projection. 



TO LAY OFF A POLAR OETHOGRAPHIQ PROJECTION 195 

meridians projected, the map may be drawn. The student 
should remember that all maps which make any claim to 
accuracy or correctness are made by locating points of 
an area to be represented according to their latitudes and 
longitudes; that is, in reference to parallels and meridians. 
It will be observed that the orthographic system of projec- 
tion crowds together areas toward the outside of the map 
and the scale of miles suitable for the central portion will 
not be correct for the outer portions. For this reason a 
scale of miles never appears upon a hemisphere made on 
this projection. 

SUMMARY 
In the orthographic projection: 

1. The eye is conceived to be at an infinite distance. 

2. Meridians and parallels are farther apart toward the center of 

the map. 

3. When a point in the equator is the center, parallels are straight 

lines. 

4. When a pole is at the center, meridians are straight lines. If the 

northern hemisphere is represented, north is not toward the top 
of the map but toward the center. 

Sterbogkaphic Projection 

In the stereographic projection the eye is conceived to be 
upon the surface of the globe, viewing the opposite hemis- 
phere. Points on the opposite hemisphere are projected 
upon a plane tangent to it. Thus in Figure 68 the eye is 
at E and sees A at A', B at B', C at C , etc. If the earth 
were transparent, we should see objects on the opposite 
half of the globe from the view point of this projection. 



196 



MAP PROJECTIONS 



How to Lay off an Equatorial Stereographic Projection. 

In Figure 68, E represents the eye at the equator, A and N 
are the poles and A' N' is the corresponding meridian of 




Fig. 68 



the projection with B', C , etc., as the points where the 
parallels cross the meridian. Taking the hne A' N' of Figure 
68 as -diameter, construct upon it a circle (see Fig. 69). 



THE POLAR STEEEOGRAPHIC PROJECTION 



197 



Divide the circumference into twenty-four equal parts 

and draw parallels as 
arcs of circles. Lay- 
off the equator and 
subdivide it the same 
as the central meridian, 
that is, the same as 
A'N' of Figure 68. 
Through the points in 
the equator, draw me- 
ridians as arcs of circles 
and the projection is 
complete. 

The Polar Stereo- 
graphic Projection is 

Fig. 69. Equatorial stereographic projection. made On the Same 

plan as the polar orthographic projection, excepting that 
the parallels have the 
distances from the pole 





Fig. 70. 



Polar stereographic 
projection. 



Fig. 71. Northern hemisphere in polar 
stereographic projection. 



that are represented by the points in A'N' of Figure 68 
(see Figs. 70, 71). 



198 



MAP PROJECTIONS 



Areas are crowded together toward the center of the 
map when made on the stereographic projection and a 
scale of miles suitable for the central portion would be too 
small for the outer portion. This projection is often used, 
however, because it is so easily laid off. 




Fig. 7a. Hemispheres in equatorial stereographic projection 
SUMMARY 

In the stereographic projection : 

1. The eye is conceived to be on the surface of the globe. 

2. Meridians and parallels are nearer together toward the center of 

the map. 

3. When a point in the equator is the center of the map, parallels and 

meridians are represented as arcs of circles. 

4. When a pole is the center, meridians are straight lines. 

Globular Projection 

With the eye at an infinite distance (as in the ortho- 
graphic projection), parallels and meridians are nearer 
together toward the outside of the map; with the eye on 
the surface (as in the stereographic projection), they are 
nearer together toward the center of the map. It would 
seem reasonable to expect that with the eye at some point 



THE POLAR STEREOGHAPHIC PROJECTION 



199 




Fig. 73 



intermediate between an infinite distance from the surface 
and the surface itself that the parallels and meridians 
■would be equidistant at dif- 
ferent portions of the map. 
That point is the sine of an 
angle of 45°, or a Kttle less 
than the length of a radius 
away from the surface. To 
find this distance at which 
the eye is conceived to be 
placed in the globular projec- 
tion, make a circle of the same 
size as the one which is the 
basis of the map to be made, 
draw two radii at an angle of 45° (one eighth of the circle) 

and draw a line, 
AB, from the ex- 
tremity of one ra- 
dius perpendicular 
to the other radius. 
The length of this 
perpendicular is the 
distance sought (A B, 
Fig. 73). 

• Thus with the eye 
at E (Fig. 74) the 
pole A is projected 
to the tangent plane 
at A', B at B', etc., 
and the distances 
A'B', B'C, etc., are 
practically equal so that they are constructed as though 
they wej-e equal in projecting parallels and meridians. 




Fig. 74 



200 



MAP PROJECTIONS 




Fig. 75. Hemispheres in equatorial globular 
projection 



How to lay off an Equatorial Globular Projection. As in 

the orthographic or stereographic projections, a circle is 

divided into equal parts, 
according to the number 
of parallels desired, the 
central meridian and 
equator being subdi- 
vided into half as many 
equal parts. Parallels 
and meridians may be 
drawn as arcs of circles, being sufficiently accurate for 
ordinary purposes (see Fig. 75). 

The polar globular projection is laid off precisely 
hke the orthographic and the stereographic projections 
having the pole as the center, excepting that the con- 
centric circles representing 
parallels are equidistant (see 
Fig. 76). 

By means of starhke addi- 
tions to the polar globular pro- 
jection (see Fig. 77), the entire 
globe ma^y be represented. If 
folded back, the rays of the 
star would meet at the south 
pole. It should be noticed 
that " south " in this pro- 
jection is in a line directly 
away from the center; that is, the top of the map is south, 
the bottom south, and the sides are also south. While 
portions of the southern hemisphere are thus spread out, 
proportional areas are well represented, South America 
and Africa being shown with little distortion of area and 
outline. 




Fig. 76. Polar globular projection 



THE POLAR GLOBULAR PROJECTION 



201 



4 



The globular projection is much used to represent 
hemispheres, or with the 
star map to represent 
the entire globe, because 
the parallels on a me- 
ridian or meridians on a 
parallel are equidistant 
and show Kttle exagger- 
ation of areas. For this 
reason it is sometimes 
called an equidistant 
projection, although 
there are other equi- 
distant projections. It 




-"WIS 
Fig. 77. World in polar globular projection 



is also called the De la Hire projection from its discoverer 
(1640-1718). 

SUMMARY 

In the globular projection; 

1. The eye is conceived to be at a certain distance from the globe 

(sine 45°). 

2. Meridians are divided equidistantly by parallels, and parallels are 

divided equidistantly by meridians. 

3. When a pole is the center of the map, meridians are straight lines. 

4. There is little distortion of areas. 

The Gnomonig Projection 

When we look up at the sky we see what appears to be 
a great dome in which the sun, moon, planets, and stars 
are located. We seem to be at the center of this celestial 
sphere, and were we to imagine stars and other heavenly 
bodies to be projected beyond the dome to an imaginary 
plane we should have a gnomonic projection. Because 
of its obvious convenience in thus showing the position 



202 



MAP PROJECTIONS 



of celestial bodies, this projection is a very old one, having 
often been used by the ancients for celestial maps. 

Since the eye is at the center for mapping the celestial 
sphere, it is conceived to be at the center of the earth in 
projecting parallels and meridians of the earth. As will 

be seen from Figure 78, 
the distortion is very 
great away from the 
center of the map and 
an entire hemisphere 
cannot be shown. 

All great circles on 
this projection are repre- 
sented as straight lines. 
This win be apparent if 
one imagines himself at 
the center of a trans- 
parent globe having par- 
allels and meridians 
traced upon it. Since 
the plane of every great 
circle passes through the 
center of the globe, the 
eye at that point wiU 
see every portion of a 
great circle as in one 
plane and will project 
As will be shown later, it is be- 
cause of this fact that sailors frequently use maps made 
on this projection. 

To Lay off a Polar Gnomonic Projection. Owing to the 
fact that parallels get so much farther apart away from 
the center of the map, the gnomonic projection is almost 




Fig. 78 



it as a straight line. 



GREAT CIRCLE SAILING 



203 




never made with any other point than the pole for center, 

and then only for latitudes about forty degrees from the 

pole. The polar gno- 

monic projection is 

made like the polar 

projections previously 

described, excepting 

that parallels intersect 

the meridians at the 

distances represented 

in. Figure 78. The 

meridians, being great 

circles, are represented 

as straight lines and 

the parallels as concen- 

. "^ . Fig. 79. Polar gnomonic projection 

trie Circles. 
Great Circle Sailing. It would seem at first thought 

that a ship sailing to a 
point due eastward, 
say from New York to 
Oporto, would follow 
the course of a paral- 
lel, that is, would sail 
due eastward. This, 
however, would not be 
its shortest course. 
The solution of the 
following httle catch 
problem in mathe- 
matical geography will 
make clear the reason 
for this. "A man was forty rods due east of a bear, 
his gun would carry only thirty rods, yet with no change 



Bear 




Fig. 80 



204 MAP PROJECTIONS 

of position he shot and Idlled the bear. Where on earth 
were they?" Solution: This could occur only -near the' 
pole where parallels are very small circles. The bear was 
westward from the man and westward is along the course 
of a parallel. The bear was thus distant forty rods in a 
curved hne from the man but the bullet flew in a straight 
Hne (see Fig. 80). 

The shortest distance between two points on the earth 
is along the arc of a great circle. A great circle passing 
through New York and Oporto passes a little to the north 
of the parallel on which both cities are located. Thus it 
is that the course of vessels plying between the United 
States and Europe curves somewhat to the northward of 
parallels. This following of a great circle by navigators 
is called great circle sailing. The equator is a great circle 
and parallels near it are almost of the same length. In 
sailing within the tropics, therefore, there is little advan- 
tage in departing from the course of a parallel. Besides 
this, the trade winds and doldrums control the choice of 
routes in that region and the Mercator projection is always , 
used in saiUng there. In higher latitudes the gnomonic 
projection is commonly used. 

Although the gnomonic projection is rarely used ex- 
cepting by sailors, it is important that students understand 
the principles underl3dng its construction since the most 
important projections yet to be discussed are based upon it. 

SUMMARY 
In the gnomonic projection: 

1. The eye is conceived to be at the center of the earth. 

2. There is great distortion of distances away from the center of the map. 

3. A hemisphere cannot be shown. 

4. All great circles are shown as straight lines. 

a. Therefore it is used largely for great circle sailing. 

5. The pole is usually the center of the map. 



H0M0L06RAPHIC PROJECTION 



205 



The Homolographic Projection 

The projections thus far discussed will not permit the 
representation of the entire globe on one ma,p, with the 
exception of the starUke extension of the polar globular 
projection. The homolographic projection is a most 
ingenious device which is used quite extensively to repre- 




Fig. 8i. Homolographic projection 

sent the entire globe without distortion of areas. It is a 
modification of the globular projection. 

How to Lay off a Homolographic Projection. First lay 
off an equatorial globular projection, omitting the parallels. 
The meridians are semi-ellipses, although those which are 
no more than 90° from the center meridian may be drawn 
as arcs of circles. 

Having laid off the meridians as in the equatorial 
globular projection, double the length of the equator, 
extending it equally in both directions, and subdivide 
these extensions as the equator was subdivided. Through 



206 



MAP PROJECTIONS 



these points of subdivision and the poles, draw ellipses for 
meridians. 

To draw the outer elliptical meridians. Set the points of 
the compasses at the distance from the point through 
which the meridian is to be drawn to the central meridian. 
Place one point of the compasses thus set at a pole and 
mark off points on the equator for foci of the ellipse. 
Drive pins in these foci and also one in a pole. Around 
these three pins form a loop with a string. Withdraw 
the pin at the pole and draw the ellipse as described on 




Fig. 82. World in homolographic projection 

page 22. This process must be repeated for each pair of 
meridians. 

The parallels are straight lines, as in the orthographic 
projection, somewhat nearer together toward the poles. 
If nine parallels are drawn on each side of the equator, 
they may be drawn in the following ratio of distances, 
beginning at the equator: 2, If, 1^, 1|, 1|, 1|, 1|, If, li 
This will give an approximately correct representation. 

One of the recent books to make frequent use of this 
projection is the " Commercial Geography " by Gannett, 
Garrison, and Houston (see Fig. 82) . 



EQUATORIAL DISTANCES OF PARALLELS 



207 



Equatorial Distances of Parallels. The following table 
gives the exact relative distances of parallels from the 
equator. Th\is if a map twenty inches wide is to be 
drawn, ten inches from equator to pole, the first parallel 
will be .69 of an inch from the equator, the second 1.37 
inches, etc. 



<p 


Dis- 
tance 





Dis- 
tance 




35° 

40 

45 


Dis- 
tance 





Dis- 
tance 


<t> 


Dis- 
tance 


<P 


Dis- 
tance 


5° 
10 
15 


.069 
.137 
.205 


20° 

25 

30 


.272 
.339 
.404 


.468 
.531 
.592 


50° 

55 

60 


.651 
.708 
.762 


65° 

70 
75 


.814 
.862 
.906 


80° 

85 

90 


.945 

.978 

1.000 



The homolographic projection is sometimes called the 
Mollweide projection from its inventor (1805), and the 
Babinet, or Babinet-homolographic projection from a 
noted cartographer who used it in an atlas (1857). From 
the fact that within any given section bounded by paral- 
lels and meridians, the area of the surface of the map is 
equal to the area within similar meridians and parallels 
of the globe, it is sometimes called the equal-surface pro- 
jection. 

SUMMARY 
In the homolographic projection: 

1. The meridians are semi-eUipses, drawn as in the globular projec- 

tion, 360° of the equator being represented. 

2. The parallels are straight lines as in the orthographic projection. 

3. Areas of the map represent equal areas of the globe. 

4. There is no distortion of area and not a very serious distortion 

of form of continents. 

5. The globe is represented as though its surface covered half of 

an exceedingly oblate spheroid. 



208 



MAP PROJECTIONS 



The Van dbr Grinten Projection 

The homolographic projection was invented early in the 
nineteenth century. At the close of the century Mr. 
Alphons Van der Grinten of Chicago invented another pro- 
jection by which the entire surface of the earth may be 
represented. This ingenious system reduces greatly the 




Fig. 83. World in Van der Grinten projection 

angular distortion incident to the homolographic projec- 
tion and for the inhabitable portions of the globe there is 
very little exaggeration of areas. 

In the Van der Grinten projection the outer boundary 
is a meridian circle, the central meridian and equator are 
straight lines, and other parallels and meridians are arcs 
of circles. The area of the circle is equal to the surface 
of a globe of one half the diameter of this circle. The 
equator is divided into 360°, but the meridians are, of 
course, divided into 180°. 



GNOMONIC CYLINDRICAL PROJECTION 



209 




Fig. 84. World in Van der Grinten projection 



A modification of 
this projection is 
shown in Figure 
83. In this the 
central meridian is 
only one half the 
length of the equa- 
tor, and parallels are 
at uniform distances 
along this meridian. 

Cylindrical Pro- 
jections 

Gnomonic Cylin- 
drical Projection. In 

this projection the sheet on which the map is to be made 

^^.,~- '— — ,^ is conceived to be 

wrapped as a cyl- 
inder around the 
globe, touching the 
equator. The eye 
is conceived to be 
at the center of the 
globe, projecting the 
parallels and meri- 
dians upon the tan- 
gent cylinder. Fig- 
ure 85 shows the 
cylinder partly un- 
wrapped with meri- 
dians as parallel 
straight' lines and 
As in the gnomonic 




Fig. 8s 



parallels also as parallel straight lines. 



JO. MATH. O£0. — 14 



210 



MAP PROJECTIONS 



projection, the parallels are increasingly farther apart away 
from the equator. 

An examination of Figure 86 will show the necessity 
for the increasing distances of parallels in higher latitudes. 
The eye at the center (E) sees A at A', B at B' , etc. Be- 
yond 45° from the equator the distance between parallels 
becomes very great. A'B' represents the same distance 
( 15° of latitude) as G' H', but is over twice as long on the 

map. At A' (60° 
60' N north latitude) the 
meridians of the 
globe are only half 
as far apart as they 
are at the equator, 
but they are repre- 
sented on the map 
as though they were 
just as far apart 
there as at the equa- 
tor. Because of the 
rapidly increasing 
distances of paral- 
lels, to represent 
higher latitudes than 
60° would reqmre a 
very large sheet, so 
the projection is usu- 
ally modified for a map of the earth as a whole, sometimes 
arbitrarily. 

G' H' is the distance from the equator to the first par- 
allel, and since a degree of latitude is about equal to a 
degree of longitude there, this distance may be taken 
between meridians. 



/ 

/ 

/ 
/ 
/ 
/ 
/ 

---<4 / 


A' 




45° 


N 




B' 




30° 


N 




F' 




15° 


N 






C' 


EC 


uai 


or 




^Vo^-^._" 


«• 




15° 


S 




V\^j/ 


r 




30° 


S 






J- 




40° 


s 




\ 

\ 
\ 
\ 
\ 
\ 
\ 


K 
1' 




60° 


s 





Fig. 86 



THE MERCATOR (CYLINDRICAL) PROJECTION 211 



Stereographic Cylindrical Projection. For reasons just 
given, the gnomonic or central cylindrical projection needs 
reduction to show the poles at all or any high latitudes 
without great distortion. Such a reduction is well shown 
in the stereographic projection. In this the eye is con- 
ceived to be on the equator, projecting each meridian from 
the view point of the meridian opposite to it. Figure 87 
shows the plan on which it is laid off, meridians being par- 
allel straight Hnes 
and equidistant and 
parallels being paral- 
lel straight hnes at 
increasing distances 
away from the equa- 
tor. 

The Mercator (Cyl- 
indrical) Projection, 
In the orthographic, 
stereographic, globu- 
lar, gnomonic, hom- 
olographic, and Van 
der Grinten projec- 
tions, parallels or 
meridians, or both, 
are represented as 
curved Hnes. It 
should be borne in 
mind that directions on the earth are determined from 
parallels and meridians. North and south are along a 
meridian and when a meridian is represented as a curved 
line, north and south are along that curved Une. Thus the 
two arrows shown at the top of Figure 81, are pointing in 
almost exactly opposite directions and yet each is point- 



/ 
/ 

• 


P- 




75° 








a' 




60° 








7^ y^ 


C 




4S' 






/^ 






30" 






A:> 


'_-'■'' d\ 


D- 




/5° 








--'""n 


F' 


fq 


ua< 


or 




r'^^^^ 


.__^__ c 


c 










\ N^^^^v 


^ ~~"'~^- / 












\\> 














"-■^IJ 












\ "s 












\ 
\ 
\ 
\ 













Fig. 87 



212 MAP PROJECTIONS 

ing due north. The arrows at the bottom point opposite 
each other, yet both point due south. The arrows point- 
ing to the right point the same way, yet one points north 
and the other points south. A Une pointing toward the 
top of a map may or may not point north. Similarly, 
parallels lie in a due east-west direction and to the right 
on a map may or may hot be to the east. 

It should be obvious by this time that the map projec- 
tions studied thus far represent directions in a most unsat- 
isfactory manner, however well they may represent areas. 
Now to the sailor the principal value of a chart is to show 
directions to steer his course by and if the direction is rep- 
resented by a curved hne it is a slow and difficult process 
for him to determine his course. We have seen that the 
gnomonic projection employs straight Unes to represent 
arcs of great circles, and, consequently, this projection is 
used in great circle sailing. The Mercator projection shows 
all parallels and meridians as straight Hues at proportional 
distances, hence directions as straight Unes, and is another, 
and the only other, kind of map used by sailors in plotting 
their courses. 

Maps in Ancient Times. Before the middle of the fif- 
teenth century, sailors did not cover very great portions 
of the earth's surface in continuous journeys out of sight 
of land where they had to be guided almost wholly by the 
stars. Mathematical accuracy in maps was not of very 
gi-eat importance in navigation imtil long journeys had to 
be made with no opportunity for verification of calcula- 
tions. Various roughly accurate map projections were 
made. The map sent to Columbus about the year 1474 
by the ItaUan astronomer Toscanelli, with which he sug- 
gested saihng directions across the " Sea of Darkness," is 
an interesting illustration of a common type of his day. 



THE MEKCATOR (CYLINDRICAL) PROJECTION 213 




214 MAP PROJECTIONS 

The long journeys of the Portuguese along the coast of 
Africa and around to Asia and the many voyages across 
the Atlantic early in the sixteenth century, made accurate 
map projection necessary. About the middle of that cen- 
tury, Emperor Charles V of Spain employed a Flemish 
mathematician named Gerhard Kramer to make maps for 
the use of his sailors. The word Kramer means, in Ger- 
man, " retail merchant," and this translated into Latin, 
then the universal language of science, becomes Mercator, 
and his invention of a very valuable and now widely used 
map projection acquired his Latinized name. 

Plan of Mercator Chart. The Mercator projection is 
made on the same plan as the other cylindrical projec- 
tions, excepting as to the distances between parallels. 
The meridians are represented as parallel Unes, whereas 
on the globe they converge. There is thus a distortion of 
longitudes, greater and greater, away from the equator. 
Now the Mercator projection makes the parallels farther 
apart away from the equator, exactly proportional to the 
meridional error. Thus at latitude 60° the meridians on 
the earth are almost exactly half as far apart as at the 
equator, but being equidistant on the map, they are rep- ■ 
resented as twice as far apart as they should be. The 
parallels in that portion of the Mercator map are accord- 
ingly made twice as far apart as they are near the equator. 
Since the distortion in latitude exactly equals the distor- 
tion in longitude and parallels and meridians are straight 
hues, all directions are represented as straight Unes. A 
navigator has simply to draw upon the map a line from 
the point where he is to the point to which he wishes to 
sail in a direct course, measure the angle which this line 
forms with a parallel or meridian, and steer his ship accord- 
ing to the bearings thus obtained. 



THE MEKCATOR (CYLINDRICAL) PROJECTION 215 



To Lay off a Mercator Projection. Figure 89 shows the 
simplest method of laying off this projection." From the 
extremity of each radius drop a hne to the nearest radius, 
parallel to the tangent A'L. The lengths of these lines, 
respectively, represent the distances* between parallels. 
Thus N'M equals CP, K' N' equals BN, A' K' equals AK. 
The meridians are 
equidistant and are 
the same distance 
apart as the first 
parallel is from the 
equator. 

The table of me- 
ridional parts on 
page 217 gives the 
relative distances of 
parallels from the 
equator. By means 
of this table a more 
exact projection 
may be laid off than 

by the method just suggested. To illustrate: Suppose we 
wish a map about twenty inches wide to include the 70th 
parallels. We find in the table that 5944.3 is the distance 
to the equator. Then, since the map is to extend 10 inches 

on each side of the equator, ^^. . „ is the scale to be used 

5944.3 

in making the map; that is, 1 inch on the map wiU be 

represented by 10 inches h- 5944.3. Suppose we wish to 

* Technically speaking, the distance is the tangent cf the angle of 
latitude and any table of natural tangents will answer nearly as well 
as the table of meridional parts, although the latter is more accurate, 
being corrected for the oblateness of the meridian. 





K 






75' 
60' 


/V 




— J 




K 






45" 


W 




/^ 


N 






30° 


/V 




1 / 


M 






15° 


w 




i 1^ 






Eau 


stor 






F C H 


F 






15° 


s 




\ J 








30' 


s 




\^ y 








ts' 


s 




^ — -^ 








60° 


S 




Fig. 8a 


L 






75° 


S 





216 



MAP PROJECTIONS 



lay off parallels ten degrees apart. The first parallel to be 
drawn north of the equator has, according to the table, 
599.1 for its meridional distance. This multiplied by 

— — — equals slightly more than 1. Hence the parallel 

10° should be laid off 1 inch from the equator. The 20th 
parallel has for its meridional distance 1217.3. This mul- 

tipHed by the scale .^..^ gives 2.03 inches from the 
5944.3 

equator. The 30th parallel has a meridional distance 




Fig. go. World in mercator projection 

1876.9, this multiplied by the scale gives 3.15 inches. In 
like manner the other parallels are laid off. The meridians 

will be i^TT—r^ X 60 or 600 inches -^ 5944.3 for every de- 
5944.3 

gree, or for ten degrees 6000 inches -^ 5944.3, which equals 

1.01 inches. This makes the map 36.36 inches long (1.01 

inches X 36 = 36.36 inches). 

We see, then, that the same scale of miles cannot be used 

for different parts of the map, though within 30° of the 

equator representations of areas will be in very nearly true 

proportions. The parallels in a map not wider than this, 

say for Africa, may be dra-\vn equidistant and the same 



THE MERCATOR (CYLINDRICAL) PROJECTION 217 



distance apart as the meridians, the inaccuracy not being 
very great. 

Table of Meridional Parts * 



1° 


59.6 


18° 


1091.1 


35° 


2231 . 1 


52° 


3647.1 


69° 


2° 


119.2 


19° 


1154.0 


36° 


2304.5 


53° 


3745.4 


70° 


3° 


178.9 


20° 


1217.3 


37° 


2378.8 


54° 


3846.1 


71° 


4° 


238.6 


21° 


1281.0 


38° 


2454 . 1 


55° 


3949.1 


72° 


5° 


298.4 


22° 


1345.1 


39° 


2530.5 


56° 


4054.9 


73° 


6° 


358.3 


23° 


1409.7 


40° 


2607.9 


57° 


4163.4 


74° 


7° 


418.3 


24° 


1474.7 


41° 


2686.5 


58° 


4274.8 


75° 


8° 


478.4 


'25° 


1540.3 


42° 


2766.3 


59° 


4389.4 


79° 


9° 


538.6 


26° 


1606.4 


43° 


2847.4 


60° 


4507.5 


77° 


10° 


599.1 


27° 


1673.1 


44° 


2929.9 


61° 


4628 . 1 


78° 


11° 


659.7 


28° 


1740.4 


45° 


3013.7 


62° 


4754.7 


79° 


12° 


720.6 


29° 


1808.3 


46° 


3099.0 


63° 


4884.5 


80° 


13° 


781.6 


30° 


1876.9 


47° 


3185.9 


64° 


5018.8 


81° 


14° 


842.9 


31° 


1946.2 


48° 


3274.5 


05° 


5158.0 


82° 


15° 


904.5 


32° 


2016.2 


49° 


3364.7 


63° 


5302.5 


83° 


16° 


966.4 


33° 


2087.0 


50° 


3456.9 


67° 


5452.8 


84° 


17° 


1028.6 


34° 


2158.6 


51° 


3551.0 


68° 


5609.5 


85° 



5773 . 1 
5944.3 
6124.0 
6313.0 
6512.4 
6723.6 
6948.1 
7187.8 
7444.8 
7722.1 
8023.1 
8352.6 
8716.4 
9122.7 
9583.0 
10114.0 
10741.7 



SUMMARY 

In the cylindrical projection: 

1. A cylinder is conceived to be wrapped around 'the globe, tangent 

to the equator. 

2. All parallels and meridians are represented as straight lines, the 

former intersecting the latter at right angles. 

3. The parallels are made at increasing distances away from the , 

equator : 

a. In the gnomonic projection, as though projected from 

the center of the earth to the tangent cylinder. 
6. In the stereographic projection, as projected from the 

equator upon an opposite meridian, the projection point 

varying for each meridian, 
c. In the Mercator projection, at distances proportional to 

the meridional excess. 
Directions are better represented in this projection than in 

any other. Here northward is directly toward the top 

of the map, eastward directly toward the right, etc. 

For this reason it is the projection most commonly 

employed for navigators' charts. 

* From Bowditch's Practical Navigator. 



218 



MAP PROJECTIONS 



4. There is great distortion of areas and outlines of continents in 

high latitudes ; Greenland appears larger than South America. 

5. The entire globe may be represented in one continuous map. 

6. The same scale of miles cannot be used for high latitudes that is 

used near the equator. 

Conic Projection 

The portion of a sphere between the 
planes of two parallels which are near 
together is very similar to the zone of a 
cone (see Fig. 91). Hence, if we imagine 
a paper in the form of a cone placed 
upon the globe and parallels and 
meridians projected upon this cone from 
the center of the globe, then this coni- 
cal map unrolled, we can understand 
this system. 

Along the parallel tangent to the 
cone, points on the map will correspond 
exactly to points upon the globe. Par- 
allels which are near the line of tangency will be repre- 
sented very much in the relative 
positions they occupy on the 
globe. In a narrow zone, there- 
fore, near the tangent parallel, 
there will be very httle distor- 
tion in latitudes and longitudes 
and an area mapped within the 
^zone will be very similar in form 
and area to the form and area 
as it appears upon the globe 
itself. For this reason the conic ^'^' ^ 

projection, or some modification of it, is almost always 
employed in representing small areas of the earth's surface. 




Fig. 91 




TO LAY OFF A CONIC PROJECTION 



219 



If the forty-fifth parallel 




To Lay off a Conic Projection 

is the center of the 

area to be mapped, 

draw two straight 

lines tangent to the 

forty-fifth parallel 

of a circle (see Fig. 

93). Project upon 

these lines points 

for parallels as in 

the gnomonic pro- 
jection. With the 

apex as center, 

draw arcs of circles 

through these points for parallels. Meridians are straight 

lines meeting at the 
apex and are equidis- 
tant along any parallel. 
It will be observed 
that parallels are farther 
apart away from the 
tangent parallel (45°, 
in this case) as in the 
Mercator projection 
they are farther apart 
away from the equator, 
which is tangent to the 
globe in that projection. 
There is also an exag- 
geration of longitudes 
away from the tangent 

Fig. 94- North America in conic projection parallel. BecaUSe of this 

lengthening of parallels, meridians are sometimes curved 




220 



MAP PROJECTIONS 



inwardly to prevent too much distortion of areas. The 
need for this will be apparent if one draws parallels be- 




Fig. 95. The woild in conic projection 



yond the equator, for he will find they are longer than 
the equator itself unless meridians curve inwardly there. 

By taking the tan- 
gent parallel ten degrees 
north of the equator and 
reducing distances of 
parallels, a fan-shaped 
ma,p of the world may 
be shown. In this map 
of the world on the 
conic projection, there 
is even greater distor- 
tion of parallels south of 
the equator, but since 
meridians converge somewhat north of the equator there 
is less distortion in northern latitudes. Since most of the 
land area of the globe is in the northern hemisphere, this 



k<^^W 


f^^< 


^Wm 


mM 


^A^r/^O-d^ 


TUm-V^ 


^^s^^^^ 


~^^^^^. 


^^r~f^^=^^^ 


^Ep%V\ 


y~>~Xir~t 


rX^X^-\\ 



Fig. 96. Europe in conic projection 



INTERSECTING CONIC PEOJECTION 



221 



projection is much better suited to represent the entire 
world than the Mercator projection. 

Bonne's (Conic) Projection. This is a modification of 
the conic projection as previously described to prevent 
exaggeration of areas away from the parallel which is con- 
ceived to be touching the globe. The central meridian is 
a' straight Une and parallels are concentric equidistant 
circles. The distance between parallels is the length 
of the arc of the circle which is used as a basis for the 
projection. For ordinary purposes, the distance AB 
(Fig. 93) may be taken for each of the distances between 
parallels. 

Having laid off the central meridian and marked off the 
arcs for parallels, the true distance of the meridian on each 
parallel is laid off and the meri- 
dian is drawn through these 
points. This gives a gentle 
inward ciirve for meridians 
toward the outside of the map 
of continents. Instead of fol- 
lowing Bonne's system with 
strict accm-acy, the map maker 
sometimes makes the curve a 
httle less in lower latitudes, 
allowing a sUght exaggeration 
of areas to permit the putting 
in of more details where they 
are needed. 

Intersecting Conic Projection. 
extent in latitude is to be represented, the cone is some- 
times conceived to cut into the sphere. In this case, 
each meridian intersects the sphere at two parallels (see 
Fig. 97) and since along and near the tangent parallels 




FJg. 97 

Where a considerable 



222 MAP PROJECTIONS 

{A and B) there is Uttle distortion, this plan is better 
adapted for a map showing greater width north and south 
than is the conic projection. 

The map of Europe well illustrates this difference. 
Etirope lies between 35° and 75° north latitude. On a 
conic projection the tangent parallel would be 55°. Near 
this parallel there would be no exaggeration of areas 
but at the extreme north and south, 20° away from this 
parallel, there would be considerable distortion. If, instead, 
we make an intersecting conic projection, we should have 
the cone pass through parallels 45° and 65° and along 
these parallels there would be no distortion and no part 
of the map being more than 10° away from these lines, 
there would be very little exaggeration anywhere. 

It should be noticed that the region between the inter- 
sections of the meridians must be projected back toward 
the center of the sphere and thus be made smaller in the 
map than it appears on the globe. The central parallel 
would be too short in proportion to the rest. Since this 
area of Europe (between 45° and 65°) is the most impor- 
tant portion and should show most details, it would be a 
serious defect, from the practical map maker's point of 
view, to minify it. 

Polyconic Projection. This differs from the conic pro- 
jection in that it is readjusted at each parallel which is 
drawn, so that each one is tangent to the sphere. This 
makes the circumscribing cone bent at each paralled, a 
series of conic sections. The word polyconic means 
" many cones." The map constructed on this projection 
is thus accurate along each parallel, instead of along but 
one as in the conic projection or along two as in the inter- 
secting conic projection. For representing small areas 
this is decidedly the most accurate projection known. 



POLYCONIC PROJECTION 



223 



Since the zone along each parallel is projected on an 
independent cone, the point 
which is the apex for one cone 
will not be the same for any- 
other (unless both north and 
south latitudes are shown in 
the same map). In the conic 
projection the parallels are all 
made from the apex of the 
cone as the center. In the 
polyconic projection each par- 
allel has its own conical apex 
and hence its own center. This 
may easily be observed by a 
comparison of the parallels in 
Figure 94 (conic projection, 
air made from one center) and 
those in Figure 98 (polyconic projection, each made from 
a different center). 



^y A/^ — i — .jltair ^"QZ^ \ A \ 



Fig. g8. Africa and Europe in 
polyconic projection 



SUMMARY 



In the conic projection: 

1. A cone is conceived to be fitted about a portion of the globe, 

tangent to some parallel. 

2. The tangent parallel shows no distortion and portions near it have 

but little. This projection is therefore used extensively for 

mapping small areas. 
a. In the conic projection on the gnomonic or central plan, 
the eye is conceived to be at the center of the globe, 
parallels are crowded closer together toward the central 
parallel, and distant areas are exaggerated. 
The cone may be conceived to intersect the globe at two 
parallels, between which there is a diminution of areas and 
beyond which there is an exaggeration of areas. 



224 MAP PROJECTIONS 

6. In the Bonne projection parallels are drawn at equidistant 
intervals from a common center and meridians are slightly 
curved to prevent distortion in longitudes. 

c. In the polyconic projection many short conic sections are 
conceived to be placed about the globe, one for each parallel 
represented. Parallels are drawn from the apexes of the 



The Scale 

The area of any map bears some proportion to the actual 
area represented. If the map is so drawn that each mile 
shall be represented by one inch on the map, since one 
mile equals 63,360 inches, the scale is said to be 1 : 63360. 

This is often written fractionally, -— --rr • A scale of two 

■" 63,360 

inches to the mile is 1 : 31,680. These, of course, can be 

used only when small areas are mapped. The following 

scales wilh their equivalents are most commonly used in 

the United States Geological Survey, .the first being the 

scale employed in the valuable geological fohos covering 

a large portion of the United States. 

Scale 1:125,000, 1 mile = 0.50688 inches. 
Scale 1:90,000, 1 mile = 0.70400 inches. 
Scale 1:62,500, 1 mile = 1.01376 inches. 
Scale 1:45,000, 1 mile = 1 .40800 inches. 



Some Conclusions 

The following generaUzations from the discussion of 
map projections seem appropriate. 

1. In all maps north and south he along meridians and 
east and west along parallels. The top of the map may or 
may not be north; indeed, the cylindrical projection is the 
only one that represents meridians by perpendicular lines. 



POLYCONIC PROJECTIONS 225 

2. Maps of the same country on different projections 
may show different shapes and yet each may be correct. 
To make maps based upon some arbitrary system of 
triangles or lines is not scientific and often is not even 
helpful. 

3. Owing to necessary distortions -in projecting the 
parallels and meridians, a scale of miles can rarely be used 
with accuracy on a map showing a large area. 

4. Straight hnes on maps are not always the shortest 
distances between two points. This will be clear if we 
remember that the shortest distance between two points 
on the globe is along the arc of a great circle. Now great 
circles, such as meridians and the equator, are very often 
represented as curved lines on a map, yet along such a 
curved line is the shortest distance between any two places 
in the line on the globe which the map represents. 

5. To ascertain the scale of miles per inch used on any 
map, or verify the scale if given, measure the space along 
a meridian for one inch and ascertain as correctly as possible 
the number of degrees of latitude contained in the inch. 
Mtdtiply this by the number of miles in one degree of 
latitude, 69, and you have the number of miles on the 
earth represented by one inch on the map. 



JO. MATH. GEO. — 15 



CHAPTER XI 

THE UNITED STATES GOVERNMENT LAND SURVEY 

Allowance for Curvature. One of the best proofs that 
the earth is a sphere is the fact that in all careful measure- 
ments over any considerable area, allowance must be made 
for the curvature of the surface. If two lines be drawn 
due northward for one mile in the northern part of the 
United States or in central Europe, say from the 48th 
parallel, they will be found nearer together at the north- 
ern extremities than they are at the southern ends. 

Origin of Geometry. One of the greatest of the prac- 
tical problems of mathematics and astronomy has been 
the systematic location of Hnes and points and the measure- 
ment of surfaces of the earth by something more definite, 
more easily described and relocated than metes and bounds. 
Indeed, geometry is believed to have had its origin in the 
need of the ancient Egyptians for surveying and reloca- 
ting the boundaries of their lands after the Nile floods. 

Locating by Metes and Bounds. The system of locating 
■lands by metes and bounds prevails extensively over the 
world and, naturally enough, was followed in this country 
by the early settlers from Europe. To locate an area by 
landmarks, some point of beginning is established and the 
boundary hnes are described by means of natural objects 
such as streams, trees, well established highways,- and 
stakes, piles of stone, etc., are placed for the purpose. The 
directions are usually indicated by reference to the magnetic 
compass and distances as ascertained by surveyors' chains. 
But landmarks decay and change, and rivers change their 

226 



LOCATING BY METES AND BOUNDS 227 

courses.* The magnetic needle of the compass does not 
point due north (excepting along two or three isogonal 
lines, called agones), and varies from year to year. This 
gives rise to endless confusion, uncertainty, and litigation. 
Variation almost without limit occurs in such descrip- 
tions, and farms assume innumerable forms, sometimes 
having a score of angles. The transitory character of such 
platting of land is illustrated in the following excerpt from 
a deed to a piece of property in Massachusetts Bay Colony, 
bearing the date: " Anno Domini one thousand seven hun- 
dred and thirty-six and in the tenth year of the reign of 
our sovereign Lord George the Second, King." In this, 
Emma Blowers deeds to Wilham Stanley, " A certain par- 
cel of Upland and Swamp Ground Situate and Ijang in the 
Township of Manchester being the thirty-first lot into the 
Westerly Division of Common Rights made in said Man- 
chester by the proprietors thereof in the year of our Lord 
one thousand six hundred ninety-nine, Said lot containing 
Ten Acres, more or less, being cutted and bounded as 
foUoweth Viz: At the Northeast Corner with a maple tree 
between Sowest and Abraham Master's, from that South- 

* Where a meandering river constitutes the boundary of a nation 
or state, changes in the course of the stream give rise to problems in 
civil government, as the following incident illustrates. A minister in 
the southern part of South Dakota was called upon to officiate at a 
wedding in a home in a bend of the Missouri River. During the high 
water of the preceding spring, the river had burst over the narrow 
neck at the bend and at the time of the wedding it was flowing on 
both sides of the cut-off so that there was a doubt as to whether the 
main channel of the stream, the interstate boundary line, was north 
of them and they were in Nebraska, or south and they were still in 
South Dakota., To be assured of the legality of the marriage rite, 
the bridal couple, minister, and witnesses rowed to the north bank, 
and up on the South Dakota bluff the marriage service was per- 
formed, the bridal party returning — they cared not to which state, 
for the festivities. 



228 THE UNITED STATES GOVERNMENT LAND SURVEY 



easterly thirty-nine poles to Morgan's Stump, so called, 
from that Southeasterly fourty-four poles upon said west 
Farm Line to a black Oak tree, from that Sixty-six poles 
Northward to the first bounds, or however Otherwise the 
Said Lot is or ought to have been bounded." 

Survey of Northwest Territory. AVhen, in 1785, practi- 
cally all of the territory north and west of the Ohio River 
had -been ceded to the United States by the withdrawal of 

state claims. Congress 
provided for its sur- 
vey, profiting from the 
experiences resulting 
from hastily marked 
boundaries. Thomas 
Hutchins was ap- 
pointed Geographer of 
the United States, and 
after the selection of 
thirteen assistants, he 
was instructed to begin 
Us survey. Starting in" 
1786 from the south- 
west corner of Pennsylvania, he laid off a line due north 
to a point on the north bank of the Ohio River. From 
this point he started a line westward. According to the 
directions of Congress, every six miles along this east- 
west " geographer's line," meridians were to be laid off 
and parallels to it at intervals of six miles, each of the 
six miles square to be divided into thirty-six square 
miles and these divided into " quarters/' thus spreading a 
huge " gridiron " over the land. The larger squares were 
called " townships," an adaptation of the New England 
" town." They are commonly called " Congressional 



36 


30 


24- 


18 


12 


6 


35 


29 


23 


17 


II 


5 


34- 


28 


22 


16 


10 


4- 


33 


27 


2/ 


15 


9 


3 


32 


26 


20 


14- 


8 


2 


31 


25 


19 


13 


7' 


/ 



Fig. 99 



SURVEY OJF NORTHWEST TERRITORY 



229 



6 


5 


4- 


3 


2 


1 


7 


a 


9 


10 


II 


12 


la 


17 


16 


IS 


14- 


13 


19 


20 


2/ 


22 


23 


24- 


30 


29 


28 


27 


26 


25 


31 


32 


33 


34 


35 


36 



Fig. 100 



townships " in most parts of the United States, to distin- 
guish them from the political subdivision of the county 
called the " civil town- 
ship " or the " muni- 
cipal township." 

Jefferson is believed 
to have suggested this 
general plan which with 
some variations has 
been continued over 
the major portion of 
the United States and 
the western portion of 
Canada. Hutchinsand 
his crew laid off the 
" geographer's line " 
only forty-two miles, making seven ranges of townships 

west of the Pennsyl- 
vania state boun- 
dary, when they were 
frightened away by 
the Indians. The work 
was continued, how- 
ever, on the same gen- 
eral plan one exception 
being the method of 
numbering the sec- 
tions. In these first 
" seven ranges " the 
sections are numbered 
as in Figure 99, else- 
where in the United States they, are numbered as in 
Figure 100, and in western Canada as in Figure 101. 



31 


3Z 


33 


34 


35 


36 


30 


29 


28 


27 


26 


25 


19 


20 


21 


22 


23 


24- 


16 


17 


16 


15 


14 


13 


7 


a 


9 


10 


II 


12 


6 


5 


4- 


3 


2 


1 



Fig. loi 



230 THE UNITED STATES 60VEENMENT LAND SURVEY 

Each of the square miles is commonly called a "sec- 
tion." 

The law passed by Congress May 20, 1785, provided that, 
" The surveyors . . . shaU proceed to divide the said ter- 
ritory into townships of six miles square, by lines running 
due north and south, and others crossing these at right 
angles, as near as may be." Owing to the convergence of 
the meridians this, of coiu"se, was a mathematical impos- 
sibility; " as near as may be," however, has been broadly 
interpreted. According to the provisions of this act and 
the acts of May 18, 1796, May 10, 1800, and Feb. 11, 1805, 
and to rules of commissioners of the general land office, a 
complete system has been evolved, the main features of 
which are as follows: 

Principal Meridians. These are run due north, south, 
or north and south from some initial point selected 
with great care and located in latitude and longitude by 
astronomical means. Thirty-two or more of these prin- 
cipal meridians have been surveyed at irregular intervals 
and of varying lengths. Some of these are known by 
numbers and sonie by names. The first principal meri- 
dian is the boundary line between Indiana, and Ohio; 
the second is west of the center of Indiana, extending the 
entire length of the state; the third is in the center of 
Illinois, extending the entire length of the state; the Talla- 
hassee principal meridian passes directly through that 
city and is only about twenty-three miles long; other 
principal meridians are named Black Hills, New Mexico, 
Indian, Louisiana, Mount Diablo, San Bernardino,* etc. 

* The entire platting of the portions of the United States to which 
this discussion refers is clearly shown on the large and excellent maps 
of the United States, published by the Government and obtainable 
at the actual cost, eighty cents, from the Commissioner of the General 
Land Office, Washington, D. C. 



BASE LINES 231 

To the east, west, or east and west of principal meridians, 
north and south rows of townships called ranges are laid 
off. Each principal meridian, together with the system of 
townships based upon it, is independent of every other 
principal meridian and where two systems come together 
irregularities are foimd. 

Base Lines. Through the initial point selected from 
which to run the principal meridian, an east-west base line 
is run, at right angles to it, and corresponds to a true geo- 
graphic parallel. As ,in case of the principal meridian, 
this hne is laid off with great care since the accuracy of 
these controUing hnes determines the accuracy of the 
measurements based upon them. 

Tiers of townships are laid off and numbered north and 
south of these base lines. In locating a township the word 
tier is usually omitted; township number 4 north, range 2 
west of the Michigan principal meridian, means the town- 
ship in tier 4 north of the base line and in the second range 
west of the Michigan principal meridian. This is the 
township in which Lansing, Michigan, is located. 

The fourth principal meridian in western Illinois and 
Wisconsin has two base lines, one at its southern extremity 
extending westward to the Mississippi River and the other 
constituting the interstate boxmdary Une between Wiscon- 
sin and IlUnois. The townships of western Illinois are 
numbered from the southern base Hne, and all of those in 
Wisconsin and northeastern Minnesota are numbered from 
the northern base Une. The fourth principal meridian is 
in three sections, being divided by an eastern bend of the 
Mississippi River and by the western portion of Lake 
Superior. 

The largest area embraced within one system is that 
based upon the fifth principal meridian. This meridian 



232 THE UNITED STATES GOVERNMENT LAND SURVEY 




STANDARD PARALLELS 233 

extends northward from the mouth of the Arkansas River 
imtil it again intersects the Mississippi River in north- 
eastern Missouri and then again it appears in the big east- 
ern bend of the Mississippi River in eastern Iowa. Its 
base Une passes a few miles south of Little Rock, Arkan- 
sas, from which fact it is sometimes called the Little Rock 
base line. From this meridian and base line all of Arkan- 
sas, Missouri, Iowa, North Dakota,' and the major portions 
of Minnesota and South Dakota have been surveyed, an 
area considerably larger than that of Germany and Great 
Britain and Ireland combined. The most northern tier 
from this base lies about a mile south of the forty-ninth 
parallel, the boimdary hne between the United States and 
Canada, and is numbered 163. The southern row of sec- 
tions of tier 164 with odd lottings lies between tier 163 and 
Canada. Its most northern township is in the extreme 
northern portion of Minnesota, west of the Lake of the 
Woods, and is numbered 168. It thus hes somewhat 
more than a thousand miles north of the base from which 
it was surveyed. There are nineteen tiers south of the 
base hne in Arkansas, making the extreme length of this 
area about 1122 miles. The most eastern range from the 
fifth principal meridian is numbered 17 and its most 
western, 104, making an extent in longitude of 726 
miles. 

Standard Parallels. The eastern and western botmdaries 
of townships are, as nearly as may be, true meridians, and 
when they have been extended northward through several 
tiers, their convergence becomes considerable. At latitude 
40° the convergence is about 6.7 feet per mile or somewhat 
more than 40 feet to each township. To prevent this 
dimunition in size of townships to the north of the base 
line, standard parallels are run, along which six-mile 



234 THE UNITED STATES GOVERNMENT LAND SURVEY 



measurements are made tor a new set of townships. These 
Hnes are also called correction lines for obvious reasons. 

Division of Dakotas. When Dakota Territory was 
divided and permitted to enter the Union as two states, 
the dividing line agreed upon was the seventh standard par- 
allel from the base line of the fifth principal meridian. 
This hne is about four miles south of the parallel 46° from 
the equator and was chosen in preference to the geo- 
graphic parallel because it was the boundary Hne between 

farms, sections, town- 
ships, and, to a consid- 
erable extent, counties. 
The boundary hne be- 
tween Minnesota and 
Iowa is what is called 
a secondary base line 
and corresponds to a 
standard parallel be- 
tween tiers 100 and 101 
north of the base Hne 
of the fifth principal 
meridian. 

The standard paral- 
lels have been run at varying intervals, the present dis- 
tance being 24 miles. None at all were used in the earHer 
surveys. Since public roads are usually built on section 
and quarter section Hnes, wherever a north-south road 
crosses a correction Hne there is a " jog " in the road, as a 
glance at Figure 103 will show. 

Townships Surveyed Northward and Westward. The 
practice in surveying is to begin at the southeast corner of 
a township and measure off to the north and west. Thus 
the sections in the north and west are Hable to be larger 




Fig. I03 



LEGAL SUBDIVISIONS OF A SECTION 



235 



or smaller than 640 acres, depending upon the accuracy of 
the survey. In case of a fractional township, made by the 
intervention of large bodies of water or the meeting of 















.Ih 


rVl 


r^ 


[2 7 


4^ i 3 


1\,* 


s'-.!3r 


21/ 


* '.3r- 


2 |/ 


A^4'^ i' 


..J..^ 

5 |e- 


7 \B 


sie^ 


7 a 


5|e 


7 la 


■■^ Sivy* 

7 


SE'A 


SW'/* 


SE'A 


SWitf 


SE'/i 


swy* 


SE'A 


2 

SWi'4 


ses* 


/ 


S£>4( 


1 

■■• NW/f 

2 


NE'/* 

-r ■ 


a 


Ct 


10 


II 


12 ' 


-, SW'M 


sey* 


O 




y 




11 





1 Lot 4, Section 7 



Fig. 104 



^Lot I. Section I. 



another system of _survey or a state line, the sections bear 
the same numbers they would have if the township were 
full. Irregular surveys and other causes sometimes make 
• the townships or sec- 
tions considerably 
larger than the desired 
area. In such cases 40 
acre lots, or as near 
that size as possible, 
appear in the northern 
row " of sections, the 
other half section re- 
maining as it would 
otherwise be. These 
lots may also appear 
in the western part of 
a township, and the 

discrepancy should appear in the western half of each 
section. This is illustrated in Figure 104. 
Legal Subdivisions of a Section. The legal subdivisions 



N. '/z 
320 


Sec.4 
Acres 


N'/2 SWA- 
80 A. 


SE'/4. 
160 Acres 


Nnswyiswy^ 

ZOA. 


SE'A-SW'A 
40A. 


SW'M 

sw'y* 

SW'/4 


lOA. 



236 THE UNITED STATES GOVERNMENT LAND SURVEY 

of a section are by halves, quarters, and half quarters. The 
designation of the portions of a section is marked in Figure 
105. The abbreviations look more unintelligible than they 
really are. Thus N. E. { of S. E. J of Sec 24, T. 123 N, R. 
64 W. 5 P.M. means the northekst quarter of the southeast 
quarter of section 24, in tier of townships number 123 north, 
and in range 64 west of the fifth principal meridian. Any 
such description can easily be located on the United States 
map issued by the General Land OfSce. 



CHAPTER XII 



TBIANGULATION IN MEASUREMENT AND SURVEY 




The ability to measure the distance and size of objects 
without so much as touching them seems to the child or 
uneducated person to be a great mystery, if not an impos- 
sibility. Uninformed persons sometimes contend that 
astronomers only guess at the distances and dimensions 
of the sun, moon, or a planet The principle of such meas- 
urement is very simple and may easily be applied. 

To Measure the Width of a Stream. Suppose we wish to 
measure the width of a river, 
yard, or field without actually 
crossing it. First make a 
triangle having two equal 
sides and one -right angle 
(Fig. 106) Select some easily 
distinguished point on the 
farther side, as X (Fig. 107), and find a convenient point 
opposite it, as B. Now carry the triangle to the right or 

left of B until by sight- 

ing you see that the 

long side is in line with 
B when the short side 

is in line with X. You 

^c will then form the tri- 

angle BAX or BCX. 
It is apparent (by simi- 
lar triangles) that ^ S or CB equals BX. Measure off .AB 
or BC and you will have BX, the distance sought. If 

237 



Tig. io6 



River y or I Fi&ld 

/- I ^ 

/ I 



,^.— 



Fig 107 



238 TKIANGULATION IN MEASUREMENT AND SURVEY 



\\\ 



you measure both to the right and to the left and take the 
average of the two you will get a more nearly correct 
result. 

To Measure the Height of an Object. In a similar man- 
ner one may measure the height of a flagstaff or building. 
Let X represent the highest point in the flagstaff (Fig. 108) 
and place the triangle on or near the ground, with the 
short side toward X and long side level. The distance to 
the foot of the pole is its height. It is easy to see from 
this that if we did not have a triangle just as described, 

say the angle at the 
point of sighting was 
less, by measuring 
that angle and look- 
ing up the value of 
its tangent in a trigo- 
nometrical table, one 
could as easily cal- 
culate the height or 
distance. The angle 
of the triangle from 
which sighting was done is 45°, its tangent is 1.0000, that is, 
XB equals 1 .0000 times BC. If the angle used were 20°, in- 
stead of 45°, its tangent would be .3640; that is, XB would 
equal .3640 times BC. If the angle were 60°, the tangent 
would be 1.7321, that is, XB would equal that number 
times BC. A complete hst of tangents for whole degrees 
is given in the Appendix. With the graduated quadrant 
the student can get the noon altitude of the sun (though 
for this purpose it need not be noon), and by getting the 
length of shadow and multiplying this by its natural tan- 
gent get the height of the object. If it is a building that 
is thus measured, the distance should be measured from 







Tig. lo8 



TO MEASURE THE HEIGHT OF AN OBJECT 239 

the end of the shadow to the place directly under the point 
casting the longest shadow measured. 

Two examples may suffice to illustrate how this may 
be done. 

1. Say an object casts a shadow 100 feet from its base 
when the altitude of the sun is observed to be 58°. The 
table shows the tangent of 58° to be 1.6003. The height 
of the object, then, must be 1.6003 times 100 feet or 160.03 
feet. 

2. Suppose an object casts a shadow 100 feet when the 
sun's height is observed to be 68° 12'. Now the table does 
not give the tangent for fractions of degrees, so we must 
add to tan 68° | of the difference between the values of 
tan 68° and tan 69° (12' = ^°). 

The table shows that 

tan 68° = 2.6051, and 

tan 69° = 2.4751, hence the 
difference = 0.1300. 
I of .1300 = 0.0260, and smce 

tan 68° = 2.4751, and we have foxmd that 

tan 12' = 0.0260, it follows that 
tan 68° 12' = 2.5011. 

Multipljdng 100 feet by this number representing the 
value of tan 68° 12' 

100 feet X 2.5011 = 250.11 feet, answer. 

By simple proportion one may also measure the height 
of an object by the length of the shadow it casts. Let XB 
represent a flagstaff and BC its shadow on the ground 
(Kg. 108). Place a ten-foot pole (any other length will 
do) perpendicularly and measm-e the length of the shadow 



240 TRIANGULATION IN MEASUREMENT AND SURVEY 

it casts and immediately mark the limit of the shadow 
of the flagstaff and measure its length in a level line. 
Now the length of the flagstaff will bear the same ratio to 
the length of the pole that the length of the shadow of 
the flagstaff bears to the length of the shadow of the 
pole. If the length of the flagstaff's shadow is 60 feet and 
that of the pole is 6 feet, it is obvious that the former is 
ten times as high as the latter, or 100 feet high. In formal 
proportion 

BX:B'X'::BC:B'C'. 

To Measure the Width of the Moon. To measure the 
width of the moon if its distance is known. Cut from a 
piece of paper a circle one inch in diameter and paste it 

Moon 



Fig. 109 



high up on a window in view of the full moon. Find how 
far the eye must be placed from the disk that the face of 
the moon may be just covered by the disk. To get this 
distance it is well to have one person hold the end of a 
tapeline against the window near the disk and the observer 



HOW ASTRONOMERS MEASURE SIZES AND DISTANCES 241 

hold the line even with his eye. You then have three ele- 
ments of the following proportion : 

Dist. to disk : dist. to moon : : width of disk : width of moon. 

From these elements, multiplying extremes and means 
and dividing, it is not difficult to get the unknown element, 
the diameter of the moon. If the student is careful in his 
measurement and does not forget to reduce aU dimensions 
to the same denomination, either feet or inches, he will be 
surprised at the accuracy of his measurement, crude though 
it is. 

How Astronomers Measure Sizes and Distances. It is by 
the aid of these principles and the use of powerful and 
accurate instrmnents that the distances and dimensions of 
celestial bodies are determined, more accurately, in some 
instances, than would be Ukely to be done with rod and 
chain, were such measurement possible. 

In measuring the distance of the moon from the earth 
two observations may be made at the same moment from 
widely distant points on the earth. Thus a triangle is 
formed from station A and station B to the moon. The 
base and included angles being known, the distance can 
be calculated to the apex of the triangle, the moon. There 
are several other methods based upon the same general 
principles, such as two observations from the same point 
twelve hours apart. Since the calculations are based upon 
hues conceived to extend to the center of the earth, this is 
called the geocentric parallax (see Parallax in Glossary). 
It is impossible to get the geocentric parallax of other 
stars than the sun because they are so far away that lines 
sighted to one from opposite sides of the earth are appar- 
ently parallel. It is only by making observations six 
months apart, the diameter of the earth's orbit forming the 

JO. MATH. QBO.— 



242 TRIANGULATION IN MEASUREMENT AND SURVEY 

base of the triangle, that the parallaxes of about forty stars 
have been determined and even then the departure from 
the parallel is so exceedingly slight that the distance can 
be given only approximately. The parallax of stars is 
called hehocentric, since the base passes through the cen- 
ter of the sun. 

Survey by Triangulation 

A method very extensively employed for exact measure- 
ment of land surfaces is by laying off imaginary triangles 
across the surface, and by measuring the length of one side 
and the included angles all other dimensions may be accu- 
rately computed. Immense areas in India, Russia, and 
North America have been thus surveyed. The triangu- 
lation surveys of the United States comprise nearly a 



83140' 


87°30' 


87 


20' 


TRIANGULATION 










IN MICHIGAN 




-Triloba 






SCALE OF MILES 


X 1 


\ 






12 3 16 10 




Baldwin^ 


c / 


\\ 






„ Dexter -----^^ 
40° A----'^ 


"/ 


Morgan 


^ 


\- Mesnard 


U° 


30' -~-~-.^^^ 


F' 


— 




Carp 


30' 


87' 


40' 


87' 


30' 


87' 


20' 



Fig. no 

million square miles extending from the Atlantic to the 
Pacific. This work has been carried on by the United 
States Geological Survey for the purpose of mapping the 
topography and making geological maps, and by the United 
States Coast and Geodetic Survey. 



DETERMINATION OF BASE LINE 243 

Determination of Base Line. The surveyor selects two 
points a few miles apart where the intervening surface is 
level. The distance between these points is ascertained, 
great care being used to make it as correct as possible, for 
this is the base line and all calculations rest for their accu- 
racy upon this distance as it is the only Une measured. 
The following extracts from the Bulletin of the United 
States Geological Survey on Triangulation, No. 122, illus- 
trate the methods employed. " The Albany base Hne (in 
central Texas) is about nine miles in length and was meas- 
ured twice with a 300-foot steel tape stretched imder a 
tension of 20 pounds. The tape was supported by stakes 
at intervals of 50 feet, which were ahgned and brought to 
the grade estabUshed by more substantial supports, the 
latter having been previously set in the ground 300 feet 
apart, and upon which markings of the extremities of the 
tape were made. The two direct measurements differed 
by 0.167 foot, but when temperature corrections were 
apphed the resulting discrepancy was somewhat greater, 
owing possibly to difficulty experienced at the time of 
measurements in obtaining the true temperature of the 
tape. The adopted length of the hne after applying the 
corrections for temperature, length of tape, difference on 
posts, inchnation, sag, and sea level, was 45,793.652 feet." 
" The .base hne (near Rapid City, South Dakota) was 
measured three times with a 300-foot steel tape; temper- 
ature was taken at each tape length; the hne was supported 
at each 50 feet and was under a uniform tension of 20 
poimds. The adopted length of the line after making cor- 
rections for slope, temperature, reduction to sea level, etc., 
is 25,796.115 feet (nearly 5 miles), and the probable error 
of the three measurements is 0.84 inch." "The Gunni- 
son line (Utah) was measured under the direction of Prof. 



244 TEIANGULATION IN MEASUREMENT AND SURVEY 

A. H. Thompson, in 1875, the measurement being made 
by wooden rods carried in a trussed wooden case. These 
rods were oiled and varnished to prevent absorption of 
moisture, and their length was carefully determined by 
comparisons with standard steel rods furnished by the 
United States Coast and Geodetic Surveys." 

Completion of Triangle. From each extremity of the 
base hne a third point is sighted and with an instrument 
the angle this line forms with the base line is determined. 
Thus suppose AB (Fig. Ill) represents the base line. At 
A the angle CAB is determined and at B the angle CBA 

is determined. Then by trig- 
onometrical tables the lengths 
of lines CA and BC are ex- 
actly determined. Any one 
of these lines may now be 
used as a base for another 
triangle as with base AB. 
If the first base line is cor- 
Pig. j„ rect, and the angles are de- 

termined accurately, and 
proper allowances are made for elevations and the curva- 
ture of the earth, the measurement is very accurate and 
easily obtained, whatever the intervening obstacles between 
the points. In some places in the western part of the 
United States, long hnes, sometimes many miles in length, 
are laid off from one high elevation to another. The 
longest side thus laid off in the Rocky Mountain region 
is 183 mile long. 

" On the recent primary triangulation much of the 
observing has been done at night upon acetylene lamps; 
directions to the distant light keepers have been sent by 
the telegraphic alphabet and flashes of light, and the 




SURVEY OP INDIAN TERRITORY 245 

necessary observing towers have been biiilt by a party 
expert in that kind of work in advance of the observing 
party."* 

Survey of Indian Territory. In March, 1895, Congress 
provided for the sxirvey of the lands of Indian Territory 
and the work was placed in charge of the Director of the 
Geological Survey instead of being let out on contract as 
had been previously done. The system of running prin- 
cipal and guide naeridians, base and correction parallels, 
and township and section Unes was adopted as usual and 
since the topographic map was made under the same direc- 
tion, a survey by triangulation was made at the same time. 
The generally level character of the cotmtry made it possi- 
ble to make triangles wherever desired, so the " checker- 
board " system of townships has superimposed upon it 
triangles diagonally across the townships. In this way the 
accurate system of triangulation was used to correct the 
errors incident to a survey by the chain. Since so many 
lines were thus laid off and all were made with extreme 
accuracy, the work of making the contour map was rendered 
comparatively simple. 

* John F. Hayford, Inspector of Geodetic Work, United States 
Coast and Geodetic Survey, in a paper relating to Primary Triangu- 
lation before the Eighth International Geographic Congress, 1904. 



CHAPTER XIII 

THE EARTH IN SPACE 

The Solar System. The group of heavenly bodies to 
which the earth belongs is called, after its great central 
sun, the solar system. The members of the solar system 
are the sun; eight large planets, some having attendant 
satelhtes or moons; several himdred smaller planets called 
asteroids, or planetoids; and occasional comets and meteors. 
The planets with their satellites, and the asteroids all 
revolve around the sun in the. same direction in elliptical 
orbits not far from a common plane. Those visible to the 
naked eye may be seen not far from the ecUptic, the path 
of the sun in its apparent revolution. The comets and 
swarms of meteors also revolve around the sun in greatly 
elongated orbits. 

The solar system is widely separated from any of the 
stars, with which the planets should not be confused. If 
one could fly from the earth to the sim, 93,000,000 miles, 
in a single day, it would take him only a month to reach 
the orbit of the most distant planet, Neptune, but at that 
same terrific rate, it would take over seven htmdred years 
to reach the very nearest of the distant stars. If a circle 
three feet in diameter be made to represent the orbit of 
the earth, an object over seventy miles away would repre- 
sent the nearest of the distant stars. 

The earth's orbit as seen from the nearest star is as a 
circle a trifle over half an inch in diameter seen at a dis- 
tance of a mile. Do not imagine that the brightest stars 
are nearest. 

246 



NEBULAR HYPOTHESIS 247 

From the foregoing one should not fail to appreciate the 
inmiensity of the earth's orbit. It is small only in a rela- 
tive sense. The earth's orbit is so large that in traveling 
eighteen and one half miles the earth departs from a 
perfectly straight Hne only about one ninth of an inch; it 
is nearly 584,000,000 miles in length and the average 
orbital velocity of the earth is 66,600 miles per hour. 

Sun's Onward Motion. It has been demonstrated that 
many of the so-called fixed stars are not fixed in relation 
to each other but have " proper " motions of their own. 
It is altogether probable that each star has its own motion 
in the universe. Now the sun is simply one of the stars 
(see p. 265), and it has been demonstrated that with its 
system of planets it is moving rapidly, perhaps 40,000 
miles per hour, toward the constellation Hercules. Many 
speculations are current as to whether our sun is controlled 
by some other sun somewhat as it controls the planets, 
and also as to general star systems. Any statement of 
such conditions with present knowledge is httle, if any, 
more than a guess. 

Jfebular Hypothesis. Time was when it was considered 
impious to endeavor to ascertain the processes by which 
God works " in His mysterious way, His wonders to per- 
form;" and to assign to natural causes and conditions 
what had been attributed to God's fiat was thought sacri- 
legious. It is hoped that day has forever passed. 

This great theory as to the successive stages and con- 
ditions in the development of the solar system, while 
doubtless faulty in some details, is at present almost the 
only working hypothesis advanced and " forms the foun- 
dation of all the current speciilations on the subject." It 
gives the facts of the solar system a unity and significance 
scarcely otherwise obtainable. 



248 THE EARTH IN SPACE 

A theory or a hypothesis, if worthy of serious attention, 
is always based upon facts. Some of the facts upon 
which the nebular theory is based are as follows : 

1. All of the planets are not far from a common 
plane. 

2. They all revolve around the sun in the same direc- 
tion. 

3. Planetary rotation and revolution are in the same 
direction, excepting, perhaps, in case of Uranus and 
Neptune. 

4. The satellites revolve around their respective planets 
in the direction of their rotation and not far from the 
plane of revolution. 

5. All the members seem to be made up of the same 
kinds of material. 

6. Analogy. 

a. The nebulae we see in the heavens have the same 
general appearances this theory assumes the solar system 
to have had. 

h. The swarms of meteorites making the rings of Saturn 
are startUngly suggestive of the theory. 

c. The gaseous condition of the sun with its corona 
suggests possible earlier extensions of it. The fact that 
the sun rotates faster at its equator than at other parts 
also points toward the nebiolar theory. The contraction 
theory of the source of the sun's heat, so generally accepted, 
is a corollary of the nebular theory. 

d. The heated ~ interior of the earth and the charac- 
teristics of the geological periods suggest this theory as 
the explanation. 

The Theory. These facts reveal a system intimately 
related and pointing to a common physical cause. Accord- 
ing to the theory, at one time, countless ages ago, all 



NEBULAR HYPOTHESIS 249 

the matter now making up the solar system was in one 
great cloudhke mass extending beyond the orbit of the 
most distant planet. This matter was not distributed with 
uniform density. The greater attraction of the denser 
portions gave rise to the collection of more matter around 
them, and just as meteors striking our atmosphere gener- 
ate by friction the flash of hght, sometimes called falhng 
or shooting stars, so the clashing of particles in this nebu- 
lous mass generated intense heat. 

Rotary Motion. Gradually the whole mass balanced 
about its center of gravity and a well-defined rotary 
motion developed. As the great nebulous mass con- 
densed and contracted, it rotated faster and faster. The 
centrifugal force at the axis of rotation was, of course, 
zero and increased rapidly toward the equator. The force 
of gravitation thus being partially counteracted by cen- 
trifugal force at the equator, and less and less so at other 
points toward the axis, the mass flattened at the poles. 
The matter being so extremely thin and tenuous and acted 
upon by intense heat, also a centrifugal force, it flattened 
out more and more into a diskUke form. 

As the heat escaped, the mass contracted and rotated 
faster than ever, the centrifugal force in the outer portion 
thus increased at a greater rate than did the power of 
gravitation due to its lessening diameter. Hence, a time 
came when the centrifugal force of the outer portions 
exactly balanced the attractive power of gravitation and 
the rim or outer fragments ceased to contract toward the 
central mass; and the rest, being nearer the center of 
gravity, shrank away from these outer portions. The 
outer ring or ringhke series of fragments, thus left off, 
continued a rotary motion around the central mass, 
remaining in essentially the same plane. 



250 THE EARTH IN SPACE 

Planets Formed from Outlying Portions. Since the 
matter in the outlying portions, as in the whole mass, was 
somewhat unevenly distributed, the parts of it consoli- 
dated. The greater masses in the outer series hastened 
by their attraction the lesser particles back of them, 
retarded those ahead of them, and thus one mass was 
formed which revolved around the parent mass and 
rotated on its axis. If this body was not too dense it 
might collect into the satelhtes or moons revolving around 
it. This process continued until nine such rings or lumps 
had been thrown off, or, rather, left off. The many small 
planets around the sun between the orbit of Mars and 
that of Jupiter were probably formed from one whose parts 
were so nearly of the same mass that no one by its pre- 
ponderating attraction could gather up all into a planet. 
The explanation of the rings of Saturn is essentially the 
same. 

Conclusion as to the Nebular Hypothesis. This theory, 
with modifications in detail, forms the basis for much of 
scientific speculation in subjects having to do with the 
earth. That it is the ultimate explanation, few will be so 
hardy as to affirm. Many questions and doubts have been 
thrown on certain phases recently but it is, in a sense, 
the point of departure for other theories which may dis- 
place it. Perhaps even the best of recent theories to 
receive the thoughtful attention of the scientific world, 
the " planetesimal hypothesis," can best be understood 
in general outline, in terms of the nebular theory. 

The Planetesimal Hypothesis. This is a new explana- 
tion of the genesis of our solar system which has been 
worked out by Professors Chamberlin and Moulton of 
the University of Chicago, and is based upon a very 
careful study of astronomical facts in the hght of maths- 



NEBULAR HYPOTHESIS 261 

matics and astrophysics. It assumes the system to have 
been evolved from a spiral nebula, similar to the most 
conmion form of nebulae observed in the heavens. It is 
supposed that the nebulous condition may have been 
caused by our sim passing so near a star that the tremen- 
dous tidal strain caused the eruptive prominences (which 
the sun shoots out at frequent intervals) to be much 
larger and more vigorous than usual, and that these, when 
projected far out, were pulled forward by the passing star 
and given a revolutionary course about the sun. The arms 
of spiral nebulae have knots of denser matter at intervals 
which are supposed to be due to special explosive impulses 
and to become the centers of accretion later. The mate- 
rial thus shot out was very hot at first, but soon cooled 
into discrete bodies or particles which moved independ- 
ently about the sun Hke planets (hence the term planetesi- 
mal). When their orbits crossed or approached each other, 
the smaller particles were gathered into the knots, and 
these ultimately grew into planets. Less than one seven- 
hundredth of the Sim was necessary to form the planets 
and satellites. 

This h3^othesis differs from the nebular hypothesis in a 
number of important particulars. The latter assumes the 
earth to have been originally in a highly heated condition, 
while tmder the planetesimal hypothesis the earth may 
have been measurably cool at the surface at all times, the 
interior heat being due to the compression caused by 
gravity. The nebular hypothesis views the atmosphere as 
the thin remnant of voluminous original gases, whereas 
the new hypothesis conceives the atmosphere to have been 
gathered gradually about as fast as consumed, and to have 
come in part from the heated interior, chiefly by volcanic 
action, and in part from outer space. The oceans, accord- 



252 THE EARTH IN SPACE 

ing to the old theory, were condensed from the great 
masses of original aqueous vapors surrounding the earth ; 
according to the new theory the water was derived from 
the same sources as the atmosphere. According to the 
planetesimal hypothesis the earth, as a whole, has been 
sohd throughout its history, and never in the molten state 
assumed in the nebular hypothesis. 

Solar System not Eternal. Of one thing we may be 
reasonably certain, the solar system is not an eternal one. 
When we endeavor to extend our thought and imagina- 
tion backward toward " the beginning," it is only toward 
creation; when forward, it is only toward eternity. 

"Thy kingdom is an everlasting kingdom. 
And thy dominion endureth throughout all generations." 

— Psalms, 145, 13. 

The Mathematical Gbogeaphy of the Planets, Moon, 

AND Sun 

The following brief sketches of the mathematical geog- 
raphy of the planets give their conditions in terms corre- 
sponding to those applied to the earth. The data and 
comparisons with the earth are only approximate. The 
more exact figures are found in the table at the end of 
the chapter. 

Striving for vividness of description occasionally restilts 
in language which impHes the possibihty of human inhab- 
itancy on other celestial bodies than the earth, or suggests 
interplanetary locomotion (see p. 305). Such conditions 
exist only in the imagination. An attempt to exclude 
astronomical facts not bearing upon the topic in hand and 
not consistent with the purpose of the study, makes nec- 
essary the omission of some of the most interesting facts. 



FORM AND DIMENSIONS 253 

For such information the student should consult an astron- 
omy. The beginner should learn the names of the planets 
in the order of their nearness to the sim. Three minutes 
repetition, with an occasional review, will fix the order: 

Mercury, Venus, Earth, Mars, Asteroids, 
Jupiter, Saturn, Uranus, Neptime. 

There are obvious advantages in the following discussion 
in not observing this sequence, taking Mars first, then 
Venus, etc. 

Mars 

Form and Dimensions. In form Mars is very similar 
to the earth, being slightly more flattened toward the 
poles. Its mean diameter is 4,200 miles, a little more 
than half the earth's. A degree of latitude near the 
equator is 36.6 miles long, getting somewhat longer toward 
the poles as in case of terrestrial latitudes. 

Mars has a little less than one third the surface of the 
earth, has one seventh the volume, weighs but one ninth 
as much, is three fourths as dense, and an object on its 
surface weighs about two fifths as much as it would here. 
A man weighing one hundred and fifty pounds on the 
earth would weigh only fifty-seven pounds on Mars, could 
jump two and one half times as high or far, and could 
throw a stone two and one half times the distance he could 
here.* A pendulum clock taken from the earth to Mars 
would lose nearly nine hours in a day as the pendulum 
would tick only about seven elevenths as fast there. A 

* He coiild not throw the stone any swifter on Mara than he could 
on the earth; gravity there being so much weaker, the stone would move 
farther before faUing to the surface. 



254 THE EARTH IN SPACE 

watch, however, would run essentially the same there as 
here. As we shall see presently, either instrument would 
have to be adjusted in order to keep Martian time as the 
day there is longer than ours. 

Rotation. Because of its well-marked surface it has 
been possible to ascertain the period of rotation of Mars 
with very great precision. Its sidereal day is 24 h. 37 m. 
22.7 s. The solar day is 39 minutes longer than our solar 
day and owing to the greater eUipticity of its orbit the 
solar days vary more in length than do ours. 

Revolution and Seasons. A year on Mars has Q68 
Martian days,* and is nearly twice as long as ours. The 
orbit is much more elUptical than that of the earth, peri- 
helion being 26,000,000 miles nearer the sun than aphelion. 
For this reason there is a marked change in the amount of 
heat received when Mars is at those two points, being 
almost one and one half times as much when in perihehon 
as when in apheUon. The northern sunamers occur when 
Mars is in apheUon, so that hemisphere has longer, cooler 
summers and shorter and warmer winters than the southern 
hemisphere. 

Northern Hemisphere Sotjthebn Hemisphere 

Spring 191 days Spring 149 days 

Summer 181 days Summer • 147 days 

Autumn 149 days Autumn 191 days 

Winter 147 days Winter 181 days 

Zones. The equator makes an angle of 24° 50' with the 
planets ecliptic (instead of 23° 27' as with us) so the change 
in seasons and zones is very similar to ours, the chmate, 
of course, being vastly different, probably very cold because 
of the rarity of the atmosphere (about the same as on our 

* Mars, by Percival Lowell. 



FORM AND DIMENSIONS 255 

highest mountains) and absence of oceans. The distance 
from the sun, too, makes a great difference in climate. 
Being about one and one half times as far as from the earth, 
the sun has an apparent diameter only two thirds as 
great and only four ninths as much heat is received over 
a similar area. 

Satellites. Mars has two satellites or moons. Since 
Mars was the god of war of the Greeks these two satelUtes 
have been given the Greek names of Deimos and Phobos, 
meaning " dread " and " terror," appropriate for " dogs 
of war." They are very small, only six or seven miles in 
diameter. Phobos is so near to Mars (3,750 miles from the 
surface) that it looks almost as large to a Martian as our 
moon does to us, although not nearly so bright. Phobos, 
being so near to Mars, has a very swift motion around the 
planet, making more than three revolutions around it 
during a single Martian day. Now om- moon travels 
aroimd the earth from west to east but only about 13° in 
a day, so because of the earth's rotation the moon rises 
in the east and sets in the west. In case of Phobos, it 
revolves faster than the planet rotates and thus rises in 
the west and sets in the east. Thus if Phobos rose in the 
west at sunset in less than three hours it would be at 
meridian height and show first quarter, in five and one 
half hours it would set in the east somewhat past the 
full, and before simrise would rise again in the west almost 
at the full again. Deimos has a sidereal period of 30.3 
hours and thus rises in the east and sets in the west, the 
period from rising to settiag being 61 hours. 

Venus 

Form and Dimensions. Venus is very nearly spherical 
and has a diameter of 7,700 miles, very nearly that of the 



256 THE EARTH IN SPACE 

earth, so its latitude and longitude are very similar to ours. 
Its surface gravity is about -j^^ that of the earth. A man 
weighing 150 pounds here would weigh 135 pounds there. 
Revolution. Venus revolves aroiuid the sun in a period 
of 225 of our days, probably rotating once on the journey, 
thus keeping essentially the same face toward the sun. 
The day, therefore, is practically the same as the year, and 
the zones are two, one of perpetual sunshine and heat and 
the other of perpetual darkness and cold. Its atmosphere 
is of nearly the same density as that of the earth. Being 
a little more than seven tenths the distance of the earth 
from the sun, that blazing orb seems to have a diameter 
nearly one and one half times as great and pours nearly 
twice as much hght and heat over a similar area. Its 
orbit is more nearly circular than that of any other 
planet. 

Jupiter 

Form and Dimensions. After Venus, this is the bright- 
est of the heavenly bodies, being immensely large and 
having very high reflecting power. Jupiter is decidedly 
oblate. Its equatorial diameter is 90,000 miles and its 
polar diameter is 84,200 miles. Degrees of latitude near 
the equator are thus nearly 785 miles long, increasing to 
over 800 miles near the pole. The area of the surface is 
122 times that of the earth, its volume 1,355, its mass or 
weight 317, and its density about one fourth. 

Surface Gravity. The weight of an object on the surface 
of Jupiter is about two and two thirds times its weight 
hero. A man weighing 150 pounds here would weigh 400 
pounds there but would find he weighed nearly 80 pounds 
more near tho pole than at the equator, gravity being so 
much more powerful there. A pendulum clock taken from 



FORM AND DIMENSIONS 257 

the earth to Jupiter would gain over nine hours in a day 
and would gain or lose appreciably in changing a single 
degree of latitude because of the oblateness of the planet. 

Rotation. The rotation of this planet is very rapid, 
occup5ang a little less than ten hours, and some portions 
seem to rotate faster than others. It seems to Tae in a 
molten or liqmd state with an extensive envelope of gases, 
eddies and currents of which move with terrific speed. 
The day there is very short as compared with ours and a 
difference of one hour in time makes a difference of over 
36° in longitude, instead of 15° as with us. Their year 
being about 10,484 of their days, their solar day is only a 
few seconds longer than their sidereal day. 

Revolution. The orbit of Jupiter is elliptical, perihelion 
being about 42,000,000 miles nearer the sun than aphelion. 
Its mean distance from the sun is 483,000,000 miles, about 
five times that of the earth. The angle its equator forms 
with its ecliptic is only 3° so there is little change in 
seasons. The vertical ray of the sun never gets more 
than 3° from the equator, and the torrid zone is 6° wide. 
The circle of illumination is never more than 3° from or 
beyond a pole so the frigid zone is only 3° wide. The 
temperate* zone is 84° wide. 

Jupiter has seven moons. 

Saturn 

Form and Dimensions. The oblateness of this planet 
is even greater than that of Jupiter, being the greatest of 

* These terms are purely relative, meaning, simply, the zone on 
Jupiter corresponding in position to the temperate zone on the earth. 
The inappropriateness of the term may be seen in the fact that Jupiter 
is intensely heated, so that its surface beneath the massive hot vapors 
surrounding it is probably molten. 

JO. MATH. GEO. — 17 



258 THE EARTH IN SPACE 

the planets. Its mean diameter is about 73,000 miles. It, 
therefore, has 768 times the volume of the earth and 84 
times the surface. Its density is the lowest of the planets, 
only about one eighth as dense as the earth. Its surface 
gravity is only slightly more than that of the earth, vary- 
ing, however, 25 per cent from pole to equator. 

Rotation. Its sidereal period of rotation is about 10 h. 
14 m., varying slightly for different portions as in case 
of Jupiter. The solar day is only a few seconds longer 
than the sidereal day. 

Revolution. Its average distance from the sun is 
866,000,000 miles, varying considerably because of its 
ellipticity. It revolves about the sun in 29.46 of our 
years, thus the annual calendar must comprise 322,777 
of the planet's days. 

The inclination of Saturn's axis makes an angle of 27° 
between the planes of its equator and its ecliptic. Thus 
the vertical ray sweeps over 54° giving that width to its 
torrid zone, 27° to the frigid, and 36° to the temperate. 
Its echptic and our ecliptic form an angle of 2.5°, so we 
always see the planet very near the sun's apparent path. 

Saturn has surrounding its equator immense disks, of 
thin, gauzelike rings, extending out nearly 50,000 miles 
from the surface. These are swarms of meteors or tiny 
moons, swinging around the planet in very nearly the 
same plane, the inner ones moving faster than the outer 
ones and being so very minute that they exert no appre- 
ciable attractive influence upon the planet. 

In addition to the rings, Saturn has ten moons. 

Uranus 

Form and Dimensions. This planet, which is barely 
visible to the unaided eye, is also decidedly oblate, nearly 



FORM AND DIMENSIONS " 259 

as much so as Saturn. Its mean diameter is given as 
from 34,900 miles to 28,500 miles. Its volume, on basis 
of the latter (and latest) figures, is 47. times that of the 
earth. Its density is very low, about three tenths that of 
the earth, and its surface gravity is about the same as 
ours at the equator, increasing somewhat toward the 
pole. 

Nothing certain is known concerning its rotation as it 
has no distinct markings upon its surface. Consequently 
we know nothing as to the axis, equator, days, calendar, 
or seasons. 

Its mean distance from the sun is 19.2 times that of the 
earth and its sidereal year 84.02 of our years. 

Uranus has four satellites swinging around the planet 
in very nearly the same plane at an angle of 82.2° to the 
plane of the orbit. They move from west to east around 
the planet, not for the same reason Phobos does about 
Mars, but probably because the axis of the planet, the 
plane of its equator, and the plane of these moons has 
been tipped 97.8° from the plane of the orbit and the 
north pole has been tipped down below or south of the 
ecliptic, becoming the south pole, and giving a backward 
rotation to the planet and to its moons. 

Neptune 

Neptune is the most distant planet from the sun, is 
probably somewhat larger than Uranus, and has about 
the same density and slightly greater surface gravity. 

Owing to the absence of definite markings nothing is 
known as to its rotation. Its one moon, Uke those of 
Uranus, moves about the planet from west to east in a 
plane at an angle of 34° 48' to its ecliptic, and its back- 
ward motion suggests a similar explanation, the inclina- 



260 THE EARTH IN SPACE 

tion of its axis is more than 90° from the plane of its 
ecliptic. 

Mercury 

This is the nearest of the planets to the sun, and as it 
never gets away from the sun more than about the width 
of forty suns (as seen from the earth), it is rarely visible 
and then only after sunset in March and April or before 
sunrise in September and October. 

Form and Dimensions. Mercury has about three 
eighths the diameter of the earth, one seventh of the sur- 
face, and one eighteenth of the volume. It probably has 
one twentieth of the mass, nine tenths of the density, 
and a little less than one third of the surface gravity. 

Rotation and Revolution. It is believed that Mercury 
rotates once on its axis during one revolution. Owing to 
its elliptical orbit it moves much more rapidly when near 
perihelion than when near aphelion, and thus the sun 
loses as compared with the average position, just as it 
does in the case of the earth, and sweeps eastward about 
23i° from its average position. When in aphelion it gains 
and sweeps westward a similar amount. This shifting 
eastward making the sun "slow" and westward making 
the sun " fast " is called libration. 

Thus there are four zones on Mercury, vastly different 
from ours, indeed, they are not zones (belts) in a terres- 
trial sense. 

a. An eUiptical central zone of perpetual sunshine, 
extending from pole to pole and 133° in longitude. In 
this zone the vertical ray shifts eastward 23j° and back 
again in the short summer of about 30 days, and westward 
a similar extent during the longer winter of about 58 
days. Two and one half times as much heat is received 



ROTATION 261 

in the summer, when in perihelion, as is received in the 
winter, when in aphehon. Thus the eastward half of 
this zone has hotter summers and cooler winters than 
does the western half. Places along the eastern and 
western margin of this zone of perpetual sunshine see the 
sun on the horizon in winter and only 23^° high in the 
summer. 

b. An elliptical zone of perpetual darkness, extending 
from pole to pole and 133° wide from east to west. 

c. Two elliptical zones of alternating sunshine and 
darkness (there being practically no atmosphere on 
Mercury, there is no twilight there), each extending from 
pole to pole and 47° wide. The eastern of these zones 
has hotter summers and cooler winters than the western 
one has. 

The Moon 

Form and Dimensions. The moon is very nearly 
spherical and has a diameter of 2,163 miles, a little over 
one fourth that of the earth, its volume one forty-fourthj 
its density three fifths, its mass -g-Y^, and its surface 
gravity one sixth that upon the earth. A pendulum 
clock taken there from the earth would tick so slowly 
that it would require about sixty hours to register one of 
our days. A degree of latitude (or l9ngitude at its 
equator) is a little less than nineteen miles long. 

Rotation. The moon rotates exactly once in one revo- 
lution around the earth, that is, keeps the same face 
toward the earth, but turns different sides toward the 
sun once each month. 

Thus what we call a sidereal month is for the moon 
•itself a sidereal day, and a synodic month is its solar 
day. The latter is 29.5306 of our days, which makes the 



262 THE EARTH IN SPACE 

moon's solar day have 708 h. 44 m. 3.8 s. If its day were 
divided into twenty-four parts as is ours, each one would 
be longer than a whole day with us. 

Revolution and Seasons. The moon's orbit around the 
sun has essentially the same characteristics as to peri- 
helion, aphelion, longer and shorter days, etc., as that of 
the earth. The fact that the moon goes around the earth 
does not materially affect it from the sun's view point. 
To illustrate the moon's orbit about the sun, draw a circle 
78 inches in diameter. Make 26 equidistant dots in 
this circle to represent the earth for each new and full 
moon of the year. Now for each new moon make a dot 
one twentieth of an inch toward the center (su») from 
every other dot representing the earth, and for every full 
moon make a dot one twentieth of an inch beyond the 
alternate ones. These dots representing the moon, if 
connected, being never more than about one twentieth 
of an inch from the circle, will not vary materially from 
the circle representing the orbit of the earth, and the 
moon's orbit around the sun will be seen to have in every 
part a concave side toward the sun. 

The solar day of the moon being 29.53 of our days, its 
tropical year must contain as many of those days as that 
number is contained times in 365.25 days or about 12.4 
days. The calendar for the moon does not have any- 
thing corresponding to our month, unless each day be 
treated as a month, but has a year of 12.4 long days of 
nearly 709 hours each. The exact length of the moon's 
solar year being 12.3689 d., its calendar would have the 
peculiarity of having one leap year in every three, that is, 
two years of 12 days each and then one of 13 days, 
with an extra leap year every 28 years. 

The earth as seen from the moon is much like the moon 



ABSENCE OF ATMOSPHERE 263 

as seen from the earth, though very much larger, about 
four times as broad. Because the moon keeps the same 
face constantly toward the earth, the latter is visible to 
ofily a httle over half of the moon. On this earthward 
side our planet would be always visible, passing through 
precisely the same phases as the moon does for us, though 
in the opposite order, the time of our new moon being 
" full earth " for the moon. So brightly does our earth 
then illimainate the moon that when only the faint cres- 
cent of the sunshine is visible to us on the rim of the 
moon, we can plainly see the " earth shine " on the rest 
of the moon's surface which is toward us. 

Zones. The inclination of the plane of the moon's 
equator to the plane of the ecliptic is 1° 32' (instead of 
23° 27' as in the case of the earth). Thus its zone corre- 
sponding to our torrid* zone is 3° 4' wide, the frigid zone 
1° 32', and the temperate zones 86° 56'. 

Absence of Atmosphere. The absence of an atmosphere 
on the moon makes conditions there vastly different from 
those to which we are accustomed. Sunrise and sunset 
show no crimson tints nor beautiful coloring and there 
is no twilight. Owing to the very slow rotation of the 
moon, 709 hours from sun-noon to sun-noon, it takes 
nearly an hour for the disk of the sun to get entirely above 
the horizon on the equator, from the time .the first gUnt 
of light appears, and the time of sunset is equally pro- 
longed; as on the earth, the time occupied in rising or 
setting is longer toward the poles of the moon. The stars 

* Again we remind the reader that these terms are not appropriate 
in case of other celestial bodies than the earth. The moon has almost 
no atmosphere to retain the sun's heat during its long night of nearly 
354 hours and its dark surface must get exceedingly cold, probably 
several hundred degrees below zero. 



264 THE EARTH IN SPACE 

do not twinkle, but shine with a clear, penetrating light. 
They may be seen as easily in the daytime as at night, 
even those very near the sun. Mercury is thus visible 
the most of the time during the long daytime of 354 
hours, and Venus as well. Out of the direct rays of the 
sun, pitch darkness prevails. Thus craters of the vol- 
canoes are very dark and also cold. In the tropical portion 
the temperature probably varies from two or three hundred 
degrees below zero at night to exceedingly high tempera- 
tures in the middle of the day. During what is to the 
moon an eclipse of the sun, which occurs whenever we see 
the moon eclipsed, the sun's hght shining through our 
atmosphere makes the most beautiful of coloring as 
viewed from the moon. The moon's atmosphere is so 
rare that it is incapable of transmitting sound, so that a 
deathlike silence prevails there. Oral conversation is 
utterly impossible and the telephone and telegraph as we 
have them would be of no use whatever. Not a drop of 
water exists on that cold and cheerless satelhte. 

Perhaps it is worth noting, in conclusion, that it is 
believed that our own atmosphere is but the thin remnant 
of dense gases, and that in ages to come it will get more 
and more rarified, until at length the earth will have the 
same conditions as to temperatm-e, silence, etc., which 
now prevail on the moon. 

The Sun 

Dimensions. The diameter is 866,500 miles, nearly 
four times the distance of the moon from the earth. Its 
surface area is about 12,000 times that of the earth, and 
its volume over a milUon times. Its density is about 
one fourth that of the earth, its mass 332,000 times, and 
its surface gravity is 27.6 times our earth's. A man 



THE SUN A STAR 265 

weighing 150 pounds here would weigh over two tons 
there, his arm would be so heavy he could not raise it 
and his bony framework could not possibly support his 
body. A pendulum clock there would gain over a hun- 
dred hours in a day, so fast would the attraction of the 
sun draw the pendulum. 

Rotation. The sun rotates on its axis in about 25^ of 
our days, showing the same portion to the earth every 
27\ days. This rate varies for different portions of the 
sun, its equator rotating considerably faster than higher 
latitudes. The direction of its rotation is from west to 
east from the sun's point of view, though as viewed from 
the earth the direction is from our east to our west. The 
plane of the equator forms an angle of about 26° with the 
plane of our equator, though only about 7J° with the 
plane of the ecliptic. 

When we realize that the earth, as viewed from the sun, 
is so tiny that it receives not more than one billionth of 
its light and heat, we may form some idea of the immense 
flood of energy it constantly pours forth. 

The Sun a Star. " The word ' star ' should be omitted 
from astronomical hterature. It has no astronomic mean- 
ing. Every star visible in the most penetrating telescope 
is a hot sun. They are at all degrees of heat, from dull 
red to the most terrific white heat to which matter can be 
subjected. Leaves in a forest, from swelling bud to the 
' sere and yellow,' do not present more stages of evolution. 
A few suns that have been weighed, contain less matter 
than our own; some of equal mass; others are from ten 
to twenty and thirty times more massive, while a few are 
so immensely more massive that all hopes of comparison 
fail. 

" Every sun is in motion at great speed, due to the attrac- 



266 



THE EARTH IN SPACE 



tion and counter attraction of all the others. They go in 
every direction. Imagine the spac6 occupied by a swarm 
of bees to be magnified so that the distance between each 
bee and its neighbor should equal one hundred miles. The 
insects would fly in every possible direction of their own 

Solar System Table 



Object 


1 

S 


Mean 

Diameter 

(miles) 


Sidereal 
Day 


As compared with tlie earth* 


w5 


■s 

a 

P 


Mass 






i? 


Mercury 


? 


3,000 


88 days 


6.800 
1.900 


0.85 
0.94 


0.048 


0.330 


0.24 


0.4 


Venus 


? 


7,700 


225 days 


0.820 


0.900 


0.62 


0.7 
1.0 


Earth 


e 


7,918 


* 


1.000 

1 


1.00 


1.000 


1.000 


1.00 


Mars 


s 


4,230 


241i 37m 22.7s 


0.440 


0.73 


0.110 


0.380 


1.88 


1.5 


Jupiter 


y 


88,000 


9h5Sm 


0.040 


0.23 


317.000 


2.650 


11.86 


5.2 


Saturn 


h 


73,000 


lOh 14m 


0.010 


0.13 


95.000 


1.180 


29.46 


9.5 

■ 


Uranus 


i. 


31,700 


9 


0.003 


0.31 


14.600 


1.110 


84.02 


19.2 


Neptune 


w 


32,000 


? 


0.001 


0.34 


17.000 


1.250 
27.650 


164.78 


30.1 


Sun 





866,400 


25d 7h 48m 




0.25 


332,000.000 






Moon 


c 


2,163 


27d 7h 43m 




0.61 


0.012 


0.166 







* The dimensions of the earth and other data, are given in the 
table of geographical constants, p. 310. 



THE SUN A STAR 267 

volition. Suns move in every conceivable direction, not 
as they will, but in abject servitude to gravitation. They 
must obey the omnipresent force, and do so with mathe- 
matical accuracy." From " New Conceptions in Astron- 
omy," by Edgar L. Larkin, in Scientific American, February 
3, 1906. 



CHAPTER XIV 
historic'al sketch 

The Form of the Earth 

While various views have been held regarding the form 
of the earth, those worthy of attention* may be grouped 
under four general divisions. 

I. The Earth Flat. Doubtless the universal beHef of 
primitive man was that, save for the irregularities of moun- 
tain, hill, and valley the surface of the earth is flat. In all 
the earUest hterature that condition seems to be assumed. 
The ancient navigators could hardly have failed to observe 
the apparent convex surface of the sea and very ancient 
literature as that of Homer alludes to the bended sea. 
This, however, does not necessarily indicate a beUef in the 
spherical form of the earth. 

Although previous to his time the doctrine of the spher- 
ical form of the earth had been advanced, Herodotus 
(born about 484 B.C., died about 425 e.g.) did not believe 
in it and scouted whatever evidence was advanced in its 
favor. Thus in giving the history of the Ptolemys, kings 
of Egypt, he relates the incident of Ptolemy Necho (about 
610-595 B.C.) sending Phoenician sailors on a voyage 
around Africa, and after giving the sailors' report that 
they saw the sun to the northward of them, he says, "I, 

* As for modern, not to say recent, pseudo-scientists and alleged 
divine revealers who contend for earths of divers forms, the reader 
is referred to the entertaining chapter entitled "Some Cranks and 
their Crochets " in John Pislie's A Century of Science, also the footnote 
on pp. 267-268, Vol. I, of his Discovery of America. 

268 



THE EARTH A SPHERE 269 

for my part, do not believe them." Now seeing the sun 
to the northward is the most logical result if the earth be 
a sphere and the sailors went south of the equator or ^outh 
of the tropic of Cancer in the northern summer. 

Ancient travelers often remarked the apparent sinking 
of southern stars and rising of northern stars as they 
traveled northward, and the opposite shifting of the heav- 
ens as they traveled southward again. In traveling east- 
ward or westward there was no displacement of the heav- 
ens and travel was so slow that the difference in time of 
sunrise or star-rise could not be observed. To infer that 
the earth is curved, at least in a north-south direction, was 
most simple and logical. It is not strange that some began 
to teach that the earth is a cyhnder. Anaximander (about 
611-547 B.C.), indeed, did teach that it is a cylinder * and 
thus prepared the way for the more nearly correct theory. 

II. The Earth a Sphere. The fact that the Chaldeans 
had determined the length of the tropical year within less 
than a minute of its actual value, had discovered the pre- 
cession of the equinoxes, and could predict eclipses over 
two thousand years before the Christian era and that in 
China similar facts were known, possibly at an earher 
period, would indicate that doubtless many of the astron- 
omers of those very ancient times had correct theories as 
to the form and motions of the earth. So far as history 
has left any positive record, however, Pythagoras (about 
582-507 B.C.), a Greek f philosopher, seems to have been 
the first to advance the idea that the earth is a sphere. 
His theory being based largely upon philosophy, nothing 

* According to some authorities he taught that the earth is a 
sphere and made terrestrial and celestial globes. See Ball's History of 
Mathematics, p. 18. 

t Sometimes called a Phoenician. 



270 HISTORICAL SKETCH 

but a perfect sphere would have answered for his concep- 
tion. He was also the first to teach that the earth 
rotates * on its axis and revolves about the sun. 

Before the time of Pythagoras, Thales (about 640-546 
B.C.), and other Greek philosophers had divided the earth 
into five zones, the torrid zone being usually considered 
so fiery hot that it could not be crossed, much less inhab- 
ited. Thales is quoted by Plutarch as believing that the 
earth is a sphere, but it seems to have been proved that 
Plutarch was in error. Many of the ancient philosophers 
did not dare to teach publicly doctrines not commonly 
accepted, for fear of punishment for impiety. It is 
possible that his private teaching was different from his 
public utterances, and that after all Plutarch was right. 

Heraclitus, Plato, Eudoxus, Aristotle and many others 
in the next two centuries taught the spherical form of the 
earth, . and, perhaps, some of them its rotation. Most of 
them, however, thought it not in harmony with a perfect 
universe, or that it was impious, to consider the sun as 
predominant and so taught the geocentric theory. 

The first really scientific attempt to calculate the size 
of the earth was by Eratosthenes (about 275-195 B.C.). 
He was the keeper of the royal library at Alexandria, and 
made many astronomical measurements and calculations 
of very great value, not only for his own day but for ours 
as well. Syene, the most southerly city of the Egypt of 
his day, was situated where the sundial cast no shadow 
at the summer solstice. Measuring carefully at Alexan- 

* Strictly speaking, Pythagoras seems to have taught that both 
sun and earth revolved about a central fire and an opposite earth 
revolved about the earth as a shield from the central fire. This rather 
complicated machinery offered so many difficulties that his followers 
abandoned the idea of the central fire and "opposite earth" and had 
the earth rotate on its own axis. 



THE EARTH A SPHERE 271 

dria, he found the noon sun to be one fiftieth of the .cir- 
cumference to the south of overhead. He then multiplied 
the distance between Syene and Alexandria, 5,000 stadia, 
by 50 and got the whole circumference of the earth to be 
250,000 stadia. The distance between the cities was not 
known very accurately and his calculation probably con- 
tained a large margin of error, but the exact length of the 
Greek stadium of his day is not known* and we cannot 
tell how near the truth he came. 

Any sketch of ancient geography would be incomplete 
without mention of Strabo (about 54 B.C. — 21 a.d.) who is 
sometimes called the " father of geography." He believed 
the earth to be a sphere at the center of the universe. 
He continued the idea of the five zones, used such circles 
as had commonly been employed by astronomers and 
geographers before him, such as the equator, tropics, and 
polar circles. His work was a standard authority for 
many centuries. 

About a century after the time of Eratosthenes, Posi- 
donius, a contemporary of Strabo, made another measure- 
ment, basing his calculations upon observations of a star 
instead of the sun, and getting a smaller circumference, 
though that of Eratosthenes was probably too small. 
Strabo, Hipparchus, Ptolemy and many others made esti- 
mates as to the size of the earth, but we have no record of 
any further measurements with a view to exact calculation 
until about 814 a.d. when the Arabian caliph Al-Mamoum 
sent astronomers and surveyors northward and southward, 
carefully measuring the distance until each party found a 
star to have shifted to the south or north one degree. 

* The most reliable data seem to indicate the length of the stadium 
was 606i feet. 



272 HISTORICAL SKETCH 

This distance of two degrees was tlien miiltiplied by 180 
and the whole circumference obtained. 

The period of the dark ages was marked by a decline in 
learning and to some extent a reversion to primitive con- 
ceptions concerning the size, form, or mathematical prop- 
erties of the earth. Almost no additional knowledge 
was acquired until early in the seventeenth century. 
Perhaps this statement may appear strange to some 
readers, for this was long after the discovery of America 
by Columbus. It should be borne in mind that his voyage 
and the resulting discoveries and explorations contributed 
nothing directly to the knowledge of the form or s-ze of 
the earth. That the earth is a sphere was generally 
believed by practically all educated people for centuries 
before the days of Columbus. The Greek astronomer 
Cleomedes, writing over a thousand years before Colum- 
bus was born, said that all competent persons excepting 
the Epicureans accepted the doctrine of the spherical 
form of the earth. 

In 1615 Willebrord Snell, professor of mathematics at 
the University of Leyden, made a careful triangular sur- 
vey of the level surfaces about Leyden and calculated 
the length of a degree of latitude to be 66.73 miles. A 
recalculation of his data with corrections which he sug- 
gested gives the much more accurate measurement of 69.07 
miles. About twenty years later, an EngHshman named 
Richard Norwood made measurements and calculations 
in southern England and gave 69.5 as the length of a 
degree of latitude, the most accurate measurement up to 
that time. 

It was about 1660 when Isaac Newton (1642-1727) 
discovered the laws of gravitation, but when he applied 
the laws to the motions of the moon his calculations did 



THE EARTH AN OBLATE SPHEROID 273 

not harmonize with what he assumed to be the size 
of the earth. About 1671 the French astronomer, Jean 
Picard, by the use of the telescope, made very careful 
measurements of a little over a degree of longitude and 
obtained a close approximation to its length. Newton, 
learning of the measurement of Picard, recalculated the 
mass of the earth and motions of the moon and found his 
law of gravitation as the satisfactory explanation of all the 
conditions. Then, in 1682, after having patientlj'' waited 
over twenty years for this confirmation, he announced 
the laws of gravitation, one of the greatest discoveries 
in the history of mankind. We find in this an excellent 
instance of the interdependence of the sciences. The 
careful measurement of the size of the earth has contrib- 
uted immensely to the sciences of astronomy and physics. 

III. The Earth an Oblate Spheroid. From the many 
calculations which Newton's fertile brain could now make, 
he soon was enabled to announce that the earth must be, 
not a true sphere, but an oblate spheroid. Christian 
Huygens, a celebrated contemporary of Newton, also con- 
tended for the oblate form of the earth, although not on 
the same groimds as those advanced by Newton. 

In about 1672 the trip of the astronomer Richer to 
French Guiana, South America, and his discovery that 
pendulums swing more slowly there (see the discussion 
under the topic The Earth an Oblate Spheroid, p. 28), 
and the resulting conclusion that the earth is not a true 
sphere, but is flattened toward the poles, gave a new 
impetus to the study of the size of the earth and other 
mathematical properties of it. 

Over half a century had to pass, however, before the 
true significance of Richer's discovery was apparent to 
all or generally accepted. An instance of a commonly 

JO. MATH. GEO. — 18 



274 HISTORICAL SKETCH 

accepted reason assigned for the shorter equatorial pen- 
dulum is the following explanation which was given to 
James II of England when he made a visit to the Paris 
Observatory in 1697. " While Jupiter at times appears 
to be not perfectly spherical, we may bear in mind the 
fact that the theory of the earth being flattened is suffi- 
ciently disproven by the circular shadow which the earth 
throws on the moon. The apparent necessary shortening 
of the pendulum toward the south is really only a correc- 
tion for the expansion of the pendulum in consequence of 
the higher temperature." It is interesting to note that if 
this explanation were the true one, the average tempera- 
ture at Cayenne would have to be 43° above the boiling 
point. 

Early in the eighteenth century Giovanni Cassini, the 
astronomer in charge of the Paris Observatory, assisted by 
his son, continued the measurement begun by Picard and 
came to the conclusion that the earth is a prolate spheroid. 
A warm discussion arose and the Paris Academy of 
Sciences decided to settle the matter by careful measure- 
ments in polar and equatorial regions . 

In 1735 two expeditions were sent out, one into Lap- 
land and the other into Peru. Their measurements, while 
not without appreciable errors, showed the decided differ- 
ence of over half a mile for one degree and demonstrated 
conclusively the oblateness of a meridian and, as Voltaire 
wittily remarked at the time, " flattened the poles and 
the Cassinis." 

The calculation of the oblateness of the earth has occu- 
pied the attention of many since the time of Newton. His 
calculation was g-^^^; that is, the polar diameter was ^-g^ 
shorter than the equatorial. Huygens estimated the flat- 
tening to be about -p^. The most commonly accepted 



THE EARTH A GEOID 275 

spheroid representing the earth is the one calculated in 
1866 by A. R. Clarke, for a long time at the head of the 
English Ordnance Survey (see p. 30). Purely astronomical 
calculations, based upon the effect of the bulging of the 
equator upon the motion of the moon, seem to indicate 
slightly less oblateness than that of General Clarke. Pro- 
fessor William Harkness, formerly astronomical director 
of the United States Naval Observatory, calculated it to 
be very nearly -^1-^. 

IV. The Earth a Geoid. During recent years many 
careful measurements have been made on various portions 
of the globe and extensive pendulum tests given to ascer- 
tain the force of gravity. These measurements demon- 
strate that the earth is not a true sphere; is not an oblate 
spheroid; indeed, its figure does not correspond to that of 
any regular or symmetrical geometric form. As explained 
in Chapter II, the equator, parallels, and meridians are 
not true circles, but are more or less elhptical and wavy 
in outline. The extensive triangulation surveys and the 
apphcation of astrophysics to astronomy and geodesy 
make possible, and at the same time make imperative, a 
careful determination of the exact form of the geoid. 

The Motions of the Earth 

The Pythagoreans maintained as a principle in their 
philosophy that the earth rotates on its axis and revolves 
about the sun. Basing their theory upon a priori reason- 
ing, they had Httle better grounds for their belief than 
those who thought otherwise. Aristarchus (about 310- 
250 B.C.), a Greek astronomer, seems to have been the 
first to advance the heliocentric theory in a systematic 
.manner and one based upon careful observations and cal- 
culations. From this time, however, until the time of 



276 HISTORICAL SKETCH 

Copernicus, the geocentric theory was almost universally 
adopted. 

The geocentric theory is often called the Ptolemaic sys- 
tem from Claudius Ptolemy (not to be confused with 
ancient Egyptian kings of the same name), an Alexandrian 
astronomer and mathematician, who seems to have done 
most of his work about the middle of the second century, 
A.D. He seems to have adopted, in general, the valuable 
astronomical calculations of Hipparchus (about 180-110 
B.C.). The system is called after him because he com- 
piled so much of the observations of other astronomers 
who had preceded him and invented a most ingenious 
system of "cycles," "epicycles," "deferents," "centrics," 
and " eccentrics " (now happily swept away by the Coper- 
nican system) by which practically all of the known facts 
of the celestial bodies and their movements could be 
accounted for and yet assume the earth to be at the center 
of the universe. 

Among Ptolemy's contributions to mathematical geog- 
raphy were his employment of the latitude and longitude 
of places to represent their positions on the globe (a scheme 
probably invented by Hipparchus), and he was the first 
to use the terms " meridians of longitude " and " parallels 
of latitude." It is from the Latin translation of his sub- 
divisions of degrees that we get the terms " minutes" and 
" seconds " (for centuries the division had been followed, 
originating with the Chaldeans. See p. 141). The sixty 
subdivisions- he called first small parts; in Latin, " minu- 
tce primce," whence our term " minute." The sixty sub- 
divisions of the minute he called second small parts; in 
Latin, " minutce secundce," whence our term " second." 

The Copernican theory of the solar system, which has 
universally displaced all others, gets its name from the 



THE MOTIONS OF THE EARTH 277 

Polish astronomer Nicolas Copernicus (1473-1543). He 
revived the theory of Aristarchus, and contended that the 
earth is not at the center of the solar system, but that the 
sun is, and planets all revolve around the sun. He had 
no more reasons for this conception than for the geocentric 
theory, excepting that it violated no laws or principles, 
was in harmony with the known facts, and was simpler. 

Contemporaries and successors of Copernicus were far 
from unanimous in accepting the hehocentric theory. One 
of the dissenters of the succeeding generation is worthy of 
note for his logical though erroneous argument against it. 
Tycho Brahe * contended that the Copernican theory was 
impossible, because if the earth revolved around the sun, 
and at one season was at one side of its orbit, and at 
another was on the opposite side, the stars would appar- 
ently change their positions in relation to the earth (tech- 
nically, there would be an annual parallax), and he could 
detect no such change. His reasoning was perfectly sound, 
but was based upon an erroneous conception of the dis- 
tances of the stars. The powerful instruments of the past 
fifty years have made these parallactic motions of many of 
the stars a determinable, though a very minute, angle, and 
constitute an excellent proof of the heliocentric theory 
(see p. 109). 

Nine years after the death of Brahe, Gahleo Galilei 
(1564r-1642) by the use of his recently invented telescope 
discovered that there were moons revolving about Jupiter, 
indicating by analogy the truth of the Copernican theory. 
Following upon the heels of this came his discovery that 
Venus in its swing back and forth near the sim plainly 

* Tycho Brahe (1546-1601) a famous Swedish astronomer, was bom 
at Knudstrup, near Lund, in the south of Sweden, but spent most of 
his life in Denmark. 



278 HISTORICAL SKETCH 

shows phases just as our moon does, and appears larger 
when in the crescent than when in the full. The only 
logical conclusion was that it revolves around the sun, 
again confirming by analogy the Copernican theory. Gali- 
lei was a thorough-going Copernican in private belief, but 
was not permitted to teach the doctrine, as it was con- 
sidered unscriptural. 

As an illustration of the humiUating subterfuges to 
which he was compelled to resort in order to present an 
argument based upon the heretical theory, the following 
is a quotation from an argument he entered into con- 
cerning three comets which appeared in 1618. He based 
his argument as to their motions upon the Copernican 
system, professing to repudiate that theory at the same 
time. 

" Since the motion attributed to the earth, which I as 
a pious and Christian person consider most false, and not 
to exist, accommodates itself so well to explain so many and 
such different phenomena, I shall not feel sure that, false 
as it is, it may not just as deludingly correspond with the 
phenomena of comets.' 

One of the best supporters of this theory in the next 
generation was Kepler (1571-1630), the German astrono- 
mer, and friend and successor of Brahe. His laws of 
planetary motion (see p. 284) were, of course, based upon 
the Copernican theory, and led to Newton's discovery of 
the laws of gravitation. 

James Bradley (1693-1762) discovered in 1727 the 
aberration of light (see p. 104), and the supporters of the 
Ptolemaic sysrem were routed, logically, though more 
than a century had to pass before the heliocentric theory 
became universally accepted. 



APPENDIX 

GRAVITY 

Gravity is frequently defined as the earth's attractive 
influence for an object. Since the attractive influence 
of the mass of the earth for an object on or near its sur- 
face is lessened by centrifugal force (see p. 14) and in 
other ways (see p. 183), it is more accurate to say that 
the force of gravity is the resultant of 

a. The attractive force mutually existing between the 
earth and the object, and 

h. The lessening influence of centrifugal force due to 
the earth's rotation. 

Let us consider these two factors separately, bearing 
in mind the laws of gravitation (see p. 17). 

a. Every particle of matter attracts every other 
particle. 

(1) Hence the point of gravity for any given object 
on the surface of the earth is determined by the mass of 
the object itself as well as the mass of the earth. The 
object pulls the earth as truly and as much as the earth 
attracts the object. The common center of gravity of the 
earth and this object lie's somewhere between the center 
of the earth's mass and -the center of the mass of the 
object. Each object on the earth's surface, then, must 
have its own independent common center of gravity be- 
tween it and the center of .ha earth's mass. The position 
of this common center will vary — 

(a) As the object varies in amount of matter (first 

law), and 

279 



280 APPENDIX 

(6) As the distance of the object from the center of 
the earth's mass varies (inversely as the square of the 
distance). 

(2) Because of this principle, the position of the sun 
or moon slightly modifies the exact position of the center 
of gravity just explained. It was shown in the dis- 
cussion of tides that, although the tidal lessening of 
the weight of an object is as yet an immeasurable 
quantity, it is a calculable one and produces tides (see 
p. 183). 

h. The rotation of the earth gives a centrifugal force 
to every object on its surface, save at the poles. 

(1) Centrifugal force thus exerts a sUght Ufting influ- 
ence on objects, increasing toward the equator. This 
hghtening influence is sufficient to decrease the weight 
of an object at the equator by 2^9^ of the whole. That 
is to say, an object which weighs 288 pounds at the 
equator would weigh a poimd more if the earth did not 
rotate. Do not infer from this that the centrifugal force 
at the pole being zero, a body weighing 288 poimds at the 
equator would weigh 289 pounds at the pole, not being 
lightened by centrifugal force. This would be true if the 
earth were a sphere. The bulging at the equator decreases 
a body's weight there by j^-g as com.pared with the weight 
at the poles. Thus a body at the equator has its weight 
lessened by ^^-g because of rotation, and by ^^ because 
of greater distance from the center, or a total of jis of 
its weight as compared with its weight at the pole. A 
body weighing 195 pounds at the pole, therefore, weighs 
but 194 pounds at the equator. Manifestly the rate of 
the earth's rotation determines the amount of this cen- 
trifugal force. If the earth rotated seventeen times as 
fast, this force at the equator would exactly equal the 



GRAVITY 



281 



earth's attraction,* objects there would have no weight; 
that is, gravity would be zero. In such a case the plumb 
line at all latitudes would point directly toward the nearest 
celestial pole. A clock at the 45th parallel with a pendu- 
lum beating seconds would gain one beat every 19| minutes 
if the earth were at rest, but would lose three beats in the 
same time if the earth rotated twice as fast. 

(2) Centrifugal force due to the rotation of the earth 
not only affects the amount of gravity, but modifies the 
direction in which it is exerted. Centrifugal force acts 
in a direction at right angles to the axis, not directly 
opposite the earth's attraction excepting at the equator. 
Thus plumb hues, excepting at the equator and poles, 
are slightly tilted toward the poles. 

If the earth were at rest a plumb hne at latitude 45° 
would be in the direction toward the center of the mass 

of the earth at C (Fig. 
112). The plumb hne 
^CF would then be PC. But 
centrifugal force is ex- 
erted toward CF, and the 
resultant of the attrac- 
tion toward C and cen- 
trifugal force toward CF 
makes the Une deviate 
to a point between those 
directions, as CG, the 
true center of gravity, and 
the plumb hne becomes P'CG. The amount of the cen- 

* other things equal, centrifugal force varies with the square of 
the velocity (see p. 14), and since centrifugal force at the equator equals 
289 times gravity, if the velocity of rotation were increased 17 times, 
centrifugal force would equal gravity (17' = 289). 




Fig. lis 



282 APPENDIX 

trifugal force is so small as compared with the earth's 
attraction that this deviation is not great. It is greatest 
at the 45th parallel where it amomits to 5' 57", or nearly 
one tenth of a degree. There is an almost equal devia- 
tion due to the oblateness of the earth. At latitude 45° 
the total deviation of the plumb hne from a hne drawn 
to the center of the earth is 11' 30.65." 

LATITUDE 

In Chapter II the latitude of a place was simply defined 
as the arc of a meridian intercepted between that place 
and the equator. This is true geographical latitude, but 
the discussion of gravity places us in a position to under- 
stand astronomical and geocentric latitude, and how geo- 
graphic latitude is determined from astronomical latitude. 

Owing to the elliptical form of a meridian " circle," the 
vertex of the angle constituting the latitude of a place is 
not at the center of the globe. -A portion of a meridian 
circle near the equator is an arc of a smaller circle than 
a portion of the same meridian near the pole (see p. 43 
and Fig. 18). 

Geocentric Latitude. It is sometimes of value to speak 
of the angle formed at the center of the earth by two 
lines, one drawn to the place whose latitude is sought, 
and the other to the equator on the same meridian. This 
is called the geocentric latitude of the place. 

Astronomical Latitude. The astronomer ascertains lati- 
tude from celestial measurements by reference to a level 
line or a plumb line. Astronomical latitude, then, is the 
angle formed between the plumb line and the plane of 
the equator. 

In the discussion of gravity, the last effect of centri- 



LATITUDE 283 

fugal force noted was on the direction of the plumb line. 
It was shown that this hne, excepting at the equator and 
poles, is deviated slightly toward the pole. The effect of 
this is to increase correspondingly the astronomical lati- 
tude of a place. Thus at latitude 45°, astronomical lati- 
tude is increased by 5' 57", the amount of this deviation. 
If there were no rotation of the earth, there would be no 
deviation of the plumb line, and what we call latitude 60° 
would become 59° 54' 51". Were the earth to rotate twice 
as fast, this latitude, as determined by the same astronom- 
ical instruments, would become 60° 15' 27". 

If adjacent to a mountain, the plumb line deviates 
toward the mountain because of its attractive influence on 
the plimab bob; and other deviations are also observed, 
such as with the ebb and flow of a near by tidal wave. 
These deviations are called "station errors," and allowance 
must be made for them in making all calculations based 
upon the plumb line. 

Geographical latitude is simply the astronomical latitude, 
corrected for the deviation of the pltmib line. Were it not 
for these deviations the latitude of a place would be deter- 
mined within a few feet of perfect accuracy. As it is, 
errors of a few hundred feet sometimes may occur (see 
p. 289). 

Celestial Latitude. In. the discussion of the celestial 
sphere many circles of the celestial sphere were described 
in the same terms as circles of the earth. The celestial 
equator. Tropic of Cancer, etc., are imaginary circles which 
correspond to the terrestrial equator. Tropic of Cancer, etc. 
Now as terrestrial latitude is distance in degrees of a meri- 
dian north or south of the equator of the earth, one would 
infer that celestial latitude is the corresponding distance 
along a celestial meridian from the celestial equator, but 



284 APPENDIX 

this is not the case. Astronomers reckon celestial latitude 
from the ecliptic instead of from the celestial equator. As 
previously explained, the distance in degrees from the 
celestial equator is called declination. 

Celestial Longitude is measured in degrees along the 
ecliptic from the vernal equinox as the initial point, meas- 
ured always eastward the 360° of the ecliptic. 

In addition to the celestial pole 90° from the celestial 
equator, there is a pole of the ecliptic, 90° from the ecliptic. 
A celestial body is thus located by reference to two sets of 
circles and two poles. 

(a) Its declination from the celestial equator and posi- 
tion in relation to hour circles, as celestial meridians are 
commonly called (see Glossary). 

(6) Its celestial latitude from the ecliptic and celestial 
longitude from " ecliptic meridians." 

KEPLER'S LAWS 

These three laws find their explanation in the laws of 
gravitation, although Kepler discovered them before New- 
ton made the discovery which has immortahzed his name. 

First Law. The orbit of each planet is an ellipse, having 
the sun as a focus. 

Second Law. The planet moves about the sun at such 
rates that the straight hne connecting the center of the 
sun with the center of the planet (this hne is called the 
planet's radius vector), sweeps over equal areas in equal 
times (see Fig. 113). 

The distance of the earth's journey for each of the 
twelve months is such that the elhpse is divided into 
twelve equal areas. In the discussion of seasons we 
observed (p. 169) that when in periheUon, in January, the 



KEPLER'S LAWS 285 

earth receives more heat each day than it does when in 
aphelion, in July. The northern hemisphere, being turned 
away from the sun in January, thus has warmer winters 
than it would other-wise have, and being toward the sun 
in July, has cooler summers. This is true only for corre- 
sponding days, not for the seasons as a whole. According 
to Kepler's second law the earth must receive exactly the 
same total amount of heat from the vernal equinox (March 




P 

Perihelion 



Fig. 113 

21) to the autimanal equinox (Sept. 23), when farther from 
the sun, as from the autiminal to the vernal equinox, when 
nearer the sun. During the former period, the northern 
summer, the earth receives less heat day by day, but 
there are more days. 

Third Law. The squares of the lengths of the times (side- 
real years) of planets are proportional to the cubes of their 
distances from the sun. Thus, 

(Earth's year)'' : (Mars' year)^ : : (Earth's distance)' : 
(Mars' distance)^ Knowing the distance of the earth to 



286 APPENDIX 

the sun and the distance of a planet to the sun, we have 
three of the quantities for our proportion, calhng the 
earth's year 1, and can find the year of the planet; or, 
knowing the time of the planet, we can find its distance. 



MOTIONS OF THE EARTH'S AXIS 

■ In the chapter on seasons it was stated that excepting 
for exceedingly slow or minute changes the earth's axis 
at one time is parallel to itself at other times. There are 
three such motions of the axis. 

Precession of the Equinoxes. Since the earth is slightly 
oblate and the bulging equator is tipped at an angle of 
(23i°) to the ecliptic, the sun's attraction on this rim 
tends to draw the axis over at right angles to the equator. 
The rotation of the earth, however, tends to keep the 
axis parallel to itself, and the effect of the additional accel- 
eration of the equator is to cause the axis to rotate slowly, 
keeping the same angle to the ecliptic, however. 

At the time of Hipparchus (see p. 276), who discovered 
this rotation of the axis, the present North star. Alpha 
Ursa Minoris, was about 12° from the true pole of the 
celestial sphere, toward which the axis points. The course 
which the pole is taking is bringing it somewhat nearer 
the polestar; it is now about 1° 15' away, but a hundred 
years hence will be only half a degree from it. The period 
of this rotation is very long, about 25,000 years, or 50.2" 
each year. Ninety degrees from the ecliptic is the pole 
of the ecliptic about which the pole of the celestial equator 
rotates, and from which it is distant 23^°. 

As the axis rotates about the pole of the ecliptic, the 
point where the plane of the equator intersects the plane 



MOTIONS OP THE EARTH'S AXIS 287 

of the ecliptic, that is, the equinox, gradually shifts around 
westward. Since the vernal equinox is at a given point 
in the earth's orbit one year, and the next year is reached 
a little ahead of where it was the year before, the terra 
-precession of the equinoxes is appropriate. The sidereal 
year (see p. 132) is the time required for the earth to 
make a complete revolution in its orbit. A solar or tropi- 
cal year is the interval from one vernal equinox to the 
next vernal equinox, and since the equinoxes " precede," 
a tropical year ends about twenty minutes before the 
earth reaches the same point in its orbit a second time. 

As is shown in the discussion of the earth's revolution 
(p. 169), the earth is in perihelion December 31, making 
the northern summer longer and cooler, day by day, than 
it would otherwise be, and the winter shorter and warmer. 
The traveling of the vernal equinox around the orbit, 
however, is gradually shifting the date of perihelion, so 
that in ages yet to come perihelion will be reached in July, 
and thus terrestrial climate is gradually changing. This 
perihelion point (and with it, aphelion) has a slight west- 
ward motion of its own of 11.25" each year, making, with 
the addition of the precession of the equinoxes of 50.2", a 
total shifting of the perihelion point (see " Apsides " in 
the Glossary) of V 1.45". At this slow rate, 10,545 years 
must pass before perihelion will be reached July 1. The 
amount of the ellipticity of the earth's orbit is gradually 
decreasing, so that by the time this shifting has taken 
place the orbit wiU be so nearly circular that there may be 
but slight climatic effect of this shift of periheUon. It 
may be of interest to note that some have reasoned that 
ages ago the earth's orbit was so elliptical that the 
northern winter, occurring in aphelion, was so long and 
cold that great glaciers were formed in northern North 



288 APPENDIX 

America and Eiirope which the short, hot summers could 
not melt. The fact of the glacial age cannot be disputed, 
but this explanation is not generally accepted as satis- 
factory. 

Nutation of the Poles. Several sets of gravitative influ- 
ences cause a slight periodic motion of the earth's axis 
toward and from the pole of the ecliptic. Instead of 
" preceding " around the circle 47° in diameter, the axis 
makes a slight wavelike motion, a " nodding," as it is 
called. The principal nutatory motion of the axis is 
due to the fact that the moon's orbit about the earth 
(inclined 5° 8' to the ecliptic) glides about the ecliptic in 
18 years, 220 days, just as the earth's equator glides about 
the ecliptic once in 25,800 years. Thus through periods 
of nearly nineteen years each the obliquity of the ecliptic 
(see pp. 118, 147) gradually increases and decreases again. 
The rate of this nutation varies somewhat and is always 
very slight; at present it is 0.47" in a year. 

Wandering of the Poles. In the discussion of gravity 
(p. 279), it was shown that any change in the position of 
particles of matter effects a change in the point of gravity 
common to them. Slight changes in the crust of the earth 
are constantly taking place, not simply the gradational 
changes of wearing down mountains and building up of 
depositional features, but great diastrophic changes in 
mountain structure and continental changes of level. 
Besides these physiographic changes, meteorological con- 
ditions must be factors in displacement of masses, the 
accumulation of snow, the fluctuation in the level of great 
rivers, etc. For these reasons minute changes in the 
position of the axis of rotation must take place within the 
earth. Since 1890 such changes in the position of the axis 
within the globe have been observed and recorded. The 



MOTIONS OF THE EARTH'S AXIS 289 

"wandering of the poles,'' as this slight shifting of the 
axis is called, has been demonstrated by the variation in 
the latitudes of places. A slight increase in the latitude 
of an observatory is noticed, and at the same time a cor- 
responding decrease is observed in the latitude of an 
observatory on the opposite side of the globe. " So 
definite are the processes of practical astronomy that the 
position of the north pole can be located with no greater 
uncertainty than the area of a large Eskimo hut." * 

In 1899 the International Geodetic Association took 
steps looking to systematic and careful observations and 
records of this wandering of the poles. Four stations 
not far from the thirty-ninth parallel but widely separated 
in longitude were selected, two in the United States, one 
in Sicily, and the other in Japan. 

All of the variations since 1889 have been within an 
area less than sixty feet in diameter. 

Seven Motions of the Earth. Seven of the well-defined 
motions of the earth have been described in this book: 

1. Diurnal Rotation. 

2. Annual Revolution in relation to the sun. 

3. Monthly Revolution in relation to the moon (see 
p. 184). 

4. Precessional Rotation of Axis about the pole of the 
ecliptic. 

5. Nutation of the poles, an elliptical or wavelike motion 
in the precessional orbit of the axis. 

6. Shifting on one axis of rotation, then on another, 
leading to a " wandering of the poles." 

7. Onward motion with the whole solar system (see 
" Sun's Onward Motion," p. 247). 

* Todd's New Astronomy, p. 9^. 

JO. MATH GEO. — 19 



290 APPENDIX 

MATHEMATICAL TREATMENT OF TIDES 

The explanation of the cause of tides in the chapter 
on that subject may be relied upon in every particular, 
although mathematical details are omitted. The mathe^ 
matical treatment is difficult to make plain to those who 
have not studied higher mathematics and physics. Sim- 
plified as much as possible, it is as follows: 

Let it be borne in mind that to find the cause of tides 
we must find unbalanced forces' which change their positions. 
Surface gravity over the globe varies slightly in different 
places, being less at the equator and greater toward the 
poles. As shown elsewhere, the force of gravity at the 
equator is less for two reasons: 

a. Because of greater centrifugal force.. 

b. Because of the oblateness of the earth. 

(a) Centrifugal force being greater at the equator than 
elsewhere, there is an unbalanced force which must cause 
the waters to pile up to some extent in the equatorial 
region. If centrifugal force were sometimes greater at the 
equator and sometimes at the poles, there would be a cor- 
responding shifting of the accumulated waters and we 
should have a tide — and it would be an immense one. 
But we know that this unbalanced force does not change 
its position, and hence it cannot produce a tide. 

(6) Exactly the same course of reasoning applies to the 
unbalanced force of gravity at the equator due to its 
greater distance from the center of gravity. The position 
of this unbalanced force does not shift, and n.o tide results. 

Since the earth turns on its axis under the sun and 
moon, any unbalanced forces they may produce wiU neces- 
sarily shift as different portions of the earth are succes- 
sively turned toward or from them. Our problem, then, 



MATHEMATICAL TREATMENT OF TIDES 



291 



is to find the cause and direction of the unbalanced forces 
produced by the moon or sun. 

In Figure 114, let CA be the acceleration toward the 
moon at C, due to the moon's attraction. Let BD be 




Fig. 114 



the acceleration at B. Now B is nearer the moon than 
C, so BD will be greater than CA, since the attraction 
varies inversely as the square of the distance. 

From B construct BE equal to CA. Comparing forces 
BE and BD, the latter is greater. Completing the paral- 
lelogram, we have BFDE. Now it is a simple demonstra- 
tion in physics that if two forces act upon B, one to F and 
the other to E, the resultant of the two forces will be the 
diagonal BD. Since BE and BF combined result in BD, 
it follows that BF represents the unbalanced force at B. 

At B, then, there is an unbalanced force as compared 
with C as represented by BF. At B' the unbalanced 
force is represented by B'F'. Note the pulling direction 
in which these unbalanced forces are exerted. 

Note. — For purposes of illustration the distance of the moon 
represented in the figures is greatly diminished. The distance CA is 
taken arbitrarily, likewise the distance BD. If CA were longer, 
however, BD would be still longer; and while giving CA a different 
length would modify the form of the diagram, the mathematical rela- 
tions would remain unchanged. Because of the short distance given 
CM in the figures, the difference between the BF in Figure 114 and BF 
in Figure 11-6 is greatly exaggerated. The difference between' the 
unbalanced or tide-producing force on the side toward the moon and 
that on the opposite side is approximately .0467 BF (Fig. 114). 



292 



APPENDIX 



In Figure 113, B is farther from the moon than C, hence 
BE (equal to CA) is greater than BD, and the unbalanced 
force at B is BF, directed away from the moon. A study 
of Figures 114 and 115 will show that the unbalanced force 
on the side towards the moon {BF in Fig. 114) is slightly 
greater than the unbalanced force on the side opposite the 
moon (BF in Fig. 115). The difference, however, is ex- 




Fig. 115 



ceedingly slight, and the tide on the opposite side is prac- 
tically equal to the tide on the side toward the attracting 
body. 

Combining the arrows showing the directions of the unbal- 
anced forces in the two figures, we have the arrows shown 




-M 



Fig. ii6 



in Figure 116. The distribution and direction of the un- 
balanced forces may be thus summarized: "The disturbing 
force produces a pull along A A' and a squeeze along BB'." * 

* Mathematical Astronomy. Barlow and Bryan, p. 377. 



THE ZODIAC 293 

THE ZODIAC 

This belt in the celestial sphere is 16° wide with the 
ecliptic as the center. The width is purely arbitrary. It 
could have been wider or narrower just as weU, but was 
adopted by the ancients because the sun, moon, and plan- 
ets known to them were always seen within 8° of the path- 
way of the sun. We know now that several asteroids, as 
truly planets as the earth, are considerably farther from 
the ecliptic than 8°; indeed, Pallas is sometimes 34° from 
the ecliptic — to the north of overhead to people of north- 
ern United States or central Europe. 

Signs. As the sun " creeps backward " in the center of 
the zodiac, one revolution each year, the ancients divided 
its pathway into twelve parts, one for each month. To 
'each of these sections of thirty degrees (360° ^ 12 = 30°) 
names were assigned, all but one after animals, each one 
being considered appropriate as a " sign " of an annual 
recurrence (see p. 117). Aries seems commonly to have 
been taken as the first in the series, the beginning of 
spring. Even yet the astronomer counts the tropical year 
from the " First point of Aries," the moment the center 
of the sun crosses the celestial equator on its journey 
northward. 

As explained in the discussion of the precession of the 
equinoxes (p. 286), the point in the celestial equator where 
the center of the sun crosses it shifts westward one degree 
in about seventy years. In ancient days the First point of 
Aries was in the constellation of that name but now it is 
in the constellation to the west, Pisces. The sign Aries 
begins with the First point of Aries, and thus with the west- 
ward travel of this point all the signs have moved back 
into a constellation of a different name. Another differ- 



294 



APPENDIX 



ence between the signs and the constellations of the zodiac 
is that the star clusters are of unequal length, some more 
than 30° and some less, whereas the signs are of uniform 
length. The positions and widths of the signs and con- 



^^J^ 



-9Lo 



"-^^ 



"'^^ 



■sll 


!i 




" 


o 


M 


" 


10 






<0 


■- 


1 


Dec?2 


,, 


<-) 


U 


01 

■a 


, 


5 



\june22 



^e-/ 



-?"«,• 



Pi sees 



onst^rr 




■SsiiL 


of b. 


3s 


CM 



^«> 



rie- 



Fig. I 17 



stellations with the date when the sun enters each are 
shown in Figure 117. 

Aries, the first sign, was named after the ram, probably 
because to the ancient Chaldeans, where the name seems 
to have originated, this was the month of sacrifice. The 
sun is in Aries from March 21 until April 20. It is repre- 



THE ZODIAC 295 

sented by a small picture of a ram ( /fn?' ) or by a hiero- 
glyphic (T). 

Taurus, the second sign ( Pi$ ), was dedicated to the 
bull. In ancient times this was the first of the signs, 
the vernal equinox being at the beginning of this sign. 
According to very ancient mythology it was the bull that 
drew the sun along its " furrow " in the sky. There 
are, however, many other theories as to the origin of the 
designation. The sun is in Taiirus from April 20 until 
May 21. • 

Gemini, the third sign, signifies twins ( M ) and gets its 
name from two bright stars, Castor and Pollux, which used 
to be in this feign, but are now in the sign Cancer. The 
sun is in Gemini from May 21 until June 22. 

Cancer, the fourth sign ( !»«ig ), was named after the 
crab, probably from the fact that when in this sign the 
sun retreats back again, crabhke, toward the south. The 
sun is in Cancer from June 22 until July 23. 

Leo, signifying lion, is the fifth sign ( R|* ) and seems 
to have been adopted because the lion usually was used 
as a symbol for fire, and when the sun was in Leo the 
hottest weather occurred. The sun is in this sign from 
July 23 until August 23. 

Virgo, the virgin ( ^ ) , refers to the Chaldean myth of 
the descent of Ishtar into hades in search of her husband. 
The sun is in Virgo from August 23 imtil September 23. 

The foregoing are the summer signs and, consequently, 
the corresponding constellations are our winter constel- 
lations. It must be remembered that the sign is always 
about 30° (the extreme length of the " Dipper ") to the 
west of the constellation of the same name. 

Libra, the balances ( A ), appropriately got its name 
from the fact that the autumnal equinox, or equal balanc- 



296 APPENDIX 

ing of day and night, occurred when the sun was in the 
constellation thus named the Balances. The sun is now 
in Libra from September 23 until October 24. 

Scorpio is the eighth sign (HIE ). The scorpion was a 
s5Tnbol of darkness, and was probably used to represent 
the shortening of days and lengthening of nights. The 
sun is now in Scorpio from October 24 imtil November 23. 

Sagittarius, meaning an archer or bowman, is 
sometimes represented as a Centaur with a 
bow and arrow. The sun is in this sign from November 
23 until December 22. 

Capricorn, signifying goat, is often represented as hav- 
ing the tail of a fish ( vo^ ). It probably has its origin 
as the mythological nurse of the young solar god. The 
sun is in Capricorn from December 22 until January 20. 

Aquarius, the water-bearer ( ^ ), is the eleventh sign 
and probably has a meteorological origin, being associated 
as the cause of the winter rains of Mediterranean coun- 
tries. The sun is in this sign from January 20 imtil 
February 19. 

Pisces is the last of the twelve signs. In accordance 
with the meaning of the term, it is represented as two 
fishes ( ^ ). Its significance was probably the same as 
the water-bearer. The sun is in this sign from February 
19 until the vernal equinox, March 21, when it has com- 
pleted the "labors" of its circuit, only to begin over 
again. 

The twelve signs of the ancient Chinese zodiac were 
dedicated to a quite different set of animals; being, m 
order, the Rat, the Ox, the Tiger, the Hare, the Dragon, 
the Serpent, the Horse, the Sheep, the Monkey, the Hen,- 
the Dog, and the Pig. The Egyptians adopted with a few 
changes the signs of the Greeks. 



the zodiac 297 

Myths and Superstitions as to the Relation of 
THE Zodiac to the Earth 

When one looks at the wonders of the heavens it does 
not seem at all strange that in the early dawn of history, 
ignorance and superstition should clothe the mysterious 
luminaries of the sky with occult influences upon the 
earth, the weather, and upon human affairs. The ancients, 
observing the apparent fixity of aU the stars excepting the 
seven changing ones of the zodiac — the sun, moon, and 
five planets known to them — endowed this belt and its 
seven presiding deities with special guardianship of the 
earth, giving us seasons, with varying length of day and 
change of weather; bringing forth at its will the sprouting 
of plants and fruitage and harvest in their season; count- 
ing off inevitably the years that span human life ; bringing 
days of prosperity to some and of adversity to others: 
and marking the wars and struggles, the growth and 
decay of nations. With such a background of belief, at 
once their science and their religion, it is not strange that 
when a child was born the parents hastened to -the astrol- 
oger to learn what planet or star was in the ascendancy, 
that is, most prominent during the night, and thus learn 
in advance what his destiny would be as determined from 
the character of the star that would rule his life. 

The moon in its monthly path around the earth must 
pass through the twelve signs of the zodiac in 29^ days or 
spend about 2^ days to each sign. During the blight of 
intelligence of the dark ages, some mediaeval astrologer 
conceived the simple method of subdividing the human 
body into twelve parts to correspond to the twelve con- 
stellations of the zodiac. Beginning with the sign Aries, 
he dedicated that to th'^ 1^°°'^ +he neck he assigned to 



298 APPENDIX 

Taurus, the arms were given over to Gemini, the stars of 
Cancer were to rule the breast, the heart was presided 
over by Leo, and so on down to Pisces which was to rule 
the feet. Now anyone who was born when the moon was in 
Aries would be strong in the head, intellectual; if in Taurus, 
he would be strong in the neck and self-willed, etc. More- 
over, since the moon makes a circuit of the signs of the 
zodiac in a month, according to his simple scheme when 
the moon is in Aries the head is especially affected; then 
diseases of the head rage (or is it then that the head 
is stronger to resist disease?), and during the next few days 
when the moon is in Taurus, beware of affections of the 
neck, and so on down the list. The very simplicity of this 
scheme and ease by which it could be remembered led to 
its speedy adoption by the masses who from time imme- 
morial have sought explanations of various phenomena 
by reference to celestial bodies. 

Now there is no astronomical or geographical necessity 
for considering Aries as the first sign of the zodiac. Our 
year begins practically with the advent of the sun into 
Capricorn — the beginning of the year was made January 
1 for this very purpose. The moon is not in any peculiar 
position in relation to the earth March 21 any more than 
it is December 23. If when the calendar was revised the 
numbering of the signs of the zodiac had been changed 
also, then Capricorn, the divinities of which now rule the 
knees, would have been made to rule the head, and the 
whole artificial scheme would have been changed ! Besides, 
the sign Capricorn does not include the constellation Capri- 
corn, so with the precession of the equinoxes the subtle 
influences once assigned to the heavenly bodies of one 
constellation have been shifted to an entirely different set 
of stars! The association of storms with the sun's cross- 



THE ZODIAC 299 

ing the equinox and with the angle the cusps of the moon 
show to the observer (a purely geometric position varying 
with the position of the observer) is jn the same class as 
bad luck attending the taking up of the ashes after the sun 
has gone down or the wearing of charms against rheuma- 
tism or the " evil " eye. 

"The fault, dear Brutus, is not in our stars, 
But in ourselves, that we are underlings." 

— Shakespeare. 



800 



APPENDIX 



PRACTICAL WORK IN MATHEMATICAL 
GEOGRAPHY 

Concrete work in this subject has been suggested directly, 
by imphcation, or by suggestive queries and problems 
throughout the book. No instruments of specific char- 
acter have been suggested for use excepting such as are 
easily provided, as a graduated quadrant, compasses, an 
isosceles right triangle, etc. Interest in the subject will 
be greatly augmented if the following simple instruments, 

or similar devices, 
are made or pur- 
chased and used. 

To Make a Sundial 

This is not espe- 
cially difficult and 
may be accom- 
plished in several 
ways. A simple 
plan is shown in Figure 118. Angle BAC should be the 
co-latitude of the place, that is, the latitude subtracted 
from 90°, though this is not at all essential. The hour 
lines may be marked off according to two systems, for 
standard time or for local time. 

Standard Time Dial. If you wish your dial to indicate 
clock time as correctly as possible, it will be necessary to 
consult the analemma or an almanac to ascertain the equa- 
tion of time when the hour lines are drawn. Since the sun 
is neither fast nor slow April 14, June 15, September 1, or 
December 25, those are the easiest days on which to lay 
off the hours. On one of those dates you can lay them off 
according to a reliable timepiece. 




Fig. ii8 



PRACTICAL WORK 301 

If you mark the hour hnes at any other date, ascertain 
the equation of time (see p. 127) and make allowances 
accordingly. Suppose the date is October 27. The ana- 
lemma shows the sim to be 16 minutes fast. You should 
mark the hour lines that many minutes before the hour as 
indicated by your timepiece, that is, the noon line when 
your watch says 11:44 o'clock, the 1 o'clock line when 
the watch indicates 12:44, etc. If the equation is slow, 
say five minutes, add that time to your clock time, mark- 
ing the noon line when your watch indicates 12:05, the 
next hour line at 1 : 05, etc. It is well to begin at the hour 
for solar noon, at that time placing the board so that the 
sun's shadow is on the XII mark and after marking off the 
afternoon hours measure from the XII mark westward 
corresponding distances for the forenoon. Unless you 
chance to live upon the meridian which gives standard 
time to the belt in which you are, the noon line will be 
somewhat to the east or west of north. 

This sundial will record the apparent solar time of the 
meridian upon which the clock time is based. The differ- 
ence in the time indicated by the sundial and your watch 
at any time is the equation of time. Test the accuracy 
of your sundial by noticing the time by your watch when 
the sundial indicates noon and comparing this difference 
with the equation of time for that day. If your sundial 
is accurate, you can set your watch any clear day by look- 
ing up the equation of time and making allowances accord- 
ingly. Thus the analemma shows that on May 28 the sun 
is three minutes fast. When the sundial indicates noon 
you know it is three minutes before twelve by the clock. 

Local Time Dial. To mark the hour lines which show 
the local mean solar time (see p. 64), set the XII hour 
line due north. Note accurately the clock time when the 



302 



APPENDIX 



shadow is north. One hour later mark the shadow line 
for the I hour line, two hours later mark the II hour line, 
etc. This dial wiU indicate the apparent solar time of 
your meridian. You can set your watch by it by first 
converting it into mean solar time and then into standard 
time. (This is explained on pp. 128, 129.) 

It should be noted that these two sundials are exactly 
the same -for persons who use local time, or, living on the 
standard time meridian, use standard time. 

The Sun Boabd 

The uses of the mounted quadrant in determining 
latitude were shown in the chapter on seasons (see p. 173). 




Fig. 119 



Dr. J. Paul Goode, of the University of Chicago, has 
designed a very convenient little instrument which an- 
swers well for this and other purposes. 
A vertically placed quadrant enables one to ascertain 



PRACTICAL WORK 



303 



the altitude of the sun for determining latitude and cal- 
culating the heights of objects. 
By means of a graduated circle 
placed horizontally the azimuth 
of the sun (see Glossary) may 
be ascertained. A simple vernier 
gives the azimuth readings to 
quarter degrees. It also has a 
device for showing the area cov- 
ered by a sunbeam of a given 
size, and hence its heating power. 

The Heliodon 

This appliance was designed by 
Mr. J. F. Morse, of the MediU 
High School, Chicago. It vividly 
illustrates the apparent path of 
the sun at the equinoxes and solr 
stices at any latitude. The points ^s- "° 

of sunrise and sunset can also be shown and hence the > 
length of the longest day or night can be calculated. 




304 



APPENDIX 



WHAT KEEPS THE MEMBERS OF THE SOLAR 
SYSTEM IN THEIR ORBITS? 

When a body is thrown in a direction parallel to the 
horizon, as the bullet from a level gun, it is acted upon by 
two forces : 

(o) The projectile force of the gun, AB. (Fig. 121.) 

(&) The attractive force of the earth, AC. 

The course it will actually take from point A is the 
diagonal AA'. When it reaches A' the force AB still 



rM. 




Fig. 131 



acts (not considering the friction of the air), impelling it 
in the line A'B'. Gravity continues to pull it in the line 
A'(7, and the projectile takes the diagonal direction A' A" 
and makes the curve (not a broken line as in the figure) 
AA'A". It is obvious from this diagram that if the im- 
pelling force be sufficiently great, line AB will be so long in 
relation to line AC that the bullet will be drawn to the 
earth just enough to keep it at the same distance from 
the surface as that of its starting point. 

The amount of such a projectile force near the surface 
of the earth at the equator as would thus keep an object 



PATHS OF PROJECTILES 



305 



at an unvarying distance from the earth is 26,100 feet per 
second. Fired in a horizontal direction from a tower (not 
allowing for the friction of the air) such a bullet would 
forever circle around the earth. Dividing the circumfer- 
ence of the earth (in feet) by this number we find that 
such a bullet would return to its starting point in about 
5,000 seconds, or 1 h. 23 m., making many revolutions 
around the earth during one day. Since our greatest guns, 
throwing a ton of steel a distance of twenty-one miles, 
give their projectiles a speed of only about 2,600 feet 
per second, it will be seen that the rate we have given 
is a terrific one. If this speed were increased to 37,000 feet 



37,000 feet a second. 



Retorn of th e bullet of 26,1 QQ ft 
~l hr. 23mTaft~er. 




Fig. iM. Paths of Projectiles ol Different Velocities (Scientific American 
Supplement, Sept. 33, igoe. Reproduced by permission) 



per second, the bullet would never return to the earth. 
One is tempted here to digress and demonstrate the utter 
impossibility of human beings even " making a trip to the 
moon," to say nothing of one to a much more distant 
planet. The terrific force with which we should have to 
be hurled to get away from the earth, fourteen times the 
speed of the swiftest cannon ball, is in itself an insuperable 
difficulty. Besides this, there would have to be the most 
exact calculation of the force and direction, allowing for 
(a) the curve given a projectile by gravity, (b) the cen- 

JO. MATH. GEO, — 20 



306 APPENDIX 

trifugal force of rotation, (c) the revolution of the earth, 
(d) the revolution of the moon, (e) the friction of the air, 
a variable quantity, impossible of calculation with abso- 
lute accuracy, (/) the inevitable swerving in the air by 
reason of its currents and varying density, and (g) the 
influence on the course by the attraction of the sun and 
planets. In addition to these mathematical calculations 
as to direction and projectile force, there would be the 
problem of (h) supply of air, (i) air pressure, to which our 
bodies through the evolution of ages have become adapted, 
(j) the momentum with which we would strike into the 
moon if we did " aim " right, etc. 

Returning to our original problem, we may notice that 
if the bullet were fired horizontally at a distance of 4,000 
miles from the surface of the earth, the puli of gravity would 
be only one fourth as great (second law of gravitation), 
and the projectile would not need to take so terrific a speed 
to revolve around the earth. As we noticed in the discus- 
sion of Mars (see p. 255), the satellite Phobos is so near 
its primary, 1,600 miles from the surface, that it revolves 
at just about the rate of a cannon ball, making about three 
revolutions while the planet rotates once. 

While allusion has been made only to a bullet or a moon, 
in noticing the application of the law of projectiles, the 
principle applies equally to the planets. Governed by the 
law here illustrated, a planet will revolve about its primary 
in an orbit varying from a circle to an elongated ellipse. 
Hence we conclude that a combination of projectile and 
attractive forces keeps the members of the solar system 
in their orbits. 



PORMULAS AND TABLES 307 

FORMULAS AND TABLES 
Symbols Commonly Employed 

There are several symbols which are generally used in 
works dealing with the earth, its orbit or some of its other 
properties. To the following brief list of these are added 
a few mathematical symbols employed in this book, 
which may not be familiar to many who will use it. The 
general plan of using arbitrary symbols is shown on page 
14, where G represents universal gravitation and g repre- 
sents gravity ; C represents centrifugal force and c centri- 
fugal force due to the rotation of the earth. 

<f) (Phi), latitude. 

e (Epsilon), obliquity of the ecliptic, also eccentricity of 
an ellipse. 

IT (Pi), the nimiber which when multiplied by the diam- 
eter of a circle equals the circumference; it is 
3.14159265, nearly3.1416, nearly 3f -n-^ = 9.8696044. 

S (Delta), declination, or distance in degrees from the 
celestial equator. 

oc , " varies as; " x <x. y means x varies as y. 

<, "is less than; " x < y means x is less than y. 

>, "is greater than; " x > y means x is greater than y. 

Formulas 
The Circle and Sphere 

r = radius. c = circumference. 

d = diameter. a = area. 



808 APPENDIX 



ird = C. 
I^d. 

IT 



irr' = area. 

47rr^ = surface of sphere. 

|-7rr^ = volume of sphere = 4.18887^ (nearly). 

The Ellipse 

a = i major axis. o = oblateness. 

b = i minor axis. e = eccentricity. 

TT ah = area of ellipse. 

a — h 
= ■ 



v/^-! 



The Earth Compared with Other Bodies 

P = the radius of the body as compared with the 
radius of the earth. Thus in case of the moon, the 
moon's radius = 1081, the earth's radius = 3959, 
and P = \%l\. 

P' = surface of body as compared with that of the 
earth. 

P^ = volume of body as compared with that of the 
earth. 

TTlfl SS 

; = surface gravity as compared with that of the 
earth. 



FORMULAS AND TABLES 309 



Centrifugal Force 

c = centrifugal force, r = radius. 
V = velocity. m = mass. 

_ mv' 
r 

Lessening of surface gravity at any latitude by reason 
of the centrifugal force due to rotation. 
g = surface gravity. 

c at any latitude = -^ X cos^ </>. 

Deviation of the plumb line from true vertical by reason 
of centrifugal force due to rotation. 
d = deviation. 
d = 357" X sin 2 </>. 

Miscellaneous 

Rate of swing of pendulum varies inversely as the 

square root of the surface gravity, r =-p- 

Density of a body = — p • 

Hourly deviation of the plane of a pendulum due to 
the rotation of the earth = sin latitude X 15° {d = sin ^ 
X 15°). 

Weight of bodies above the surface of the earth. 

w = weight, 

d = distance from the center of the earth. 

Weight of bodies below the surface of the earth. 
w<x d. 



310 APPENDIX 



GEOGRAPHICAL CONSTANTS* 

Equatorial semi-axis: 

in feet 20,926,062. 

in meters 6,378,206,4 

in miles 3,963.307 

Polar semi-axis: 

in feet 20,855,121. 

in meters 6,356,583.8 

in miles 3,949.871 

Oblateness of earth 1-7-294.9784 

Circumference of equator (in miles) 24,901.96 

CSrcumference through poles (in miles) 24,859 . 76 

Area of earth's surface, square miles 196,971,984 . 

Volume of earth, cubic miles 259,944,035,515. 

Mean density (Harkness) 5 . 676 

Surface density (Harkness) 2 . 56 

Obliquity of ecliptic (see page 118) 23° 27' 4.98 a. 

Sidereal year 365 d. 6 h. 9 m. 8 . 97 s. or 365 . 25636 d. 

Tropical year 365 d. 5 h. 48 m. 45.51 s. or 365.24219 d. 

Sidereal day 23 h. 56 m. 4 . 09 s. of mean solar time. 

Distance of earth to sun, mean (in miles) 92^800,000. 

Distance of earth to moon, mean (in miles) 238,840. 

MEASURES OF LENGTH 

Statute mile. . . .- 5,280. 00 feet 

Nautical mile,t or knot 6,080. 27 

German sea mile 6,076. 22 

Prussian mile, law of 1868 24,604. 80 

Norwegian and Swedish mile 36,000. 00 

Danish mile 24,712. 51 

Russian werst, or versta 3,500. 00 

Meter 3. 28 

Fathom 6.00 

Link of surveyor's chain 0. 66 

''' Dimensions of the earth are based upon the Clarke spheroid of 
1866. 
t As defined by the United States Coast and Geodetic Survey. 



FORMULAS AND TABLES 



311 



TABLE OF NATURAL SINES AND COSINES 



Sin 




Cos 


Sin 




Cos 


Sin 




Cos 


0° 


.0000 


90° 


31° 


.5150 


59° 


61° 


.8746 


29° 


1 


.0175 


89 


32 


.5299 


58 


62 


.8829 


28 


2 


.0349 


88 


33 


.5446 


57 


63 


.8910 


27 


3 


.0523 


87 


34 


.5592 


56 


64 


.8988 


26 


4 


.0698 


86 


35 


.5736 


55 


65 


.9063 


25 


5 


.0872 


85 


36 


.5878 


54 


66 


.9135 


24 


6 


.1045 


84 


37 


.6018 


53 


67 


.9205 


23 


7 


.1219 


83 


38 


.6157 


52 


68 


.9272 


22 


8 


.1392 


82 


39 


.6293 


51 


69 


.9336 


21 


9 


.1564 


81 


40 


.6424 


50 


70 


.9397 


20 


10 


.1736 


80 


41 


.6561 


49 


71 


.9455 


19 


11 


.1908 


79 


42 


.6691 


48 


72 


.9511 


18 


12 


.2079 


78 


43 


.6820 


47 


73 


.9563 


17 


13 


.2250 


77 


44 


.6947 


46 


74 


.9613 


16 


14 


.2419 


76 


45 


.7071 


45 


75 


.9659 


15 


15 


.2588 


75 


46 


.7193 


44 


76 


.9703 


14 


16 


.2756 


74 


47 


.7314 


43 


77 


.9744 


13 


17 


.2924 


73 


48 


.7431 


42 


78 


.9781 


12 


18 


.3090 


72 


49 


.7547 


41 


79 


.9816 


11 


19 


.3256 


71 


50 


.7660 


40 


80 


.9848 


10 


20 


.3420 


70 


51 


.7771 


39 


81 


.9877 


9 


21 


.3584 


69 


52 


.7880 


38 


82 


.9903 


8 


22 


.3746 


68 


53 


.7986 


37 


83 


.9925 


7 


23 


.3907 


67 


54 


.8090 


36 


84 


.9945 


6 


24 


.4067 


66 


55 


.8192 


35 


85 


.9962 


5 


25 


.4226 


65 


56 


.8290 


34 


86 


.9976 


4 


26 


.4384 


64 


57 


.8387 


33 


87 


.9986 


3 


27 


.4540 


63 


58 


.8480 


32 


88 


.9994 


2 


28 


.4695 


62 


59 


.8572 


31 


89 


.9998 


1 


29 


.4848 


61 


60 


.8660 


30 


90 


1.0000 





30 


.5000 


60 















312 



APPENDIX 



TABLE OF NATURAL TANGENTS AND COTANGENTS 



Tan 




Cot 


Tan 




Cot 


Tan 




Cot 


0° 


.0000 


90° 


31° 


.6009 


59° 


61° 


1.8040 


29° 


1 


.0175 


89 


32 


.6249 


58 


62 


1.8807 


28 


2 


.0349 


88 


33 


.6494 


57 


63 


1.9626 


27 


3 


.0524 


87 


34 


.6745 


56 


64 


2.0503 


26 


4 


.0699 


86 


35 


.7002 


55 


65 


2.1445 


25 


5 


.0875 


85 


36 


.7265 


54 


66 


2.2460 


24 


6 


.1051 


84 


37 


.7536 


53 


67 


2.3559 


23 


7 


.1228 


83 


38 


.7813 


52 


68 


2.4751 


22 


8 


:1405 


82 


39 


.8098 


51 


69 


2.6051 


21 


9 


.1584 


81 


40 


.8391 


50 


70 


2.7475 


20 


10 


.1763 


80 


41 


.8693 


49 


71 


2 . 9042 


19 


11 


.1944 


79 


42 


.9004 


48 


72 


3.0777 


18 


12 


.2126 


78 


43 


.9325 


47 


73 


3.2709 


17 


13 


.2309 


77 


44 


.9657 


46 


74 


3.4874 


16 


14 


.2493 


76 


45 


1.0000 


45 


75 


3.7321 


15 


15 


.2679 


75 


46 


1.0355 


44 


76 


4.0108 


14 


16 


.2867 


74 


47 


1.0724 


43 


77 


4.3315 


13 


17 


.3057 


73 


48 


1.1106 


42 


78 


4.7046 


12 


18 


.3249 


72 


49 


1 . 1504 


41 


79 


5.1446 


11 


19 


.3443 


71 


50 


1.1918 


40 


80 


5.6713 


10 


20 


.3640 


70 


51 


1.2349 


39 


81 . 


6.1338 


9 


21 


.3839 


69 


52 


1.2794 


38 


82 


7.1154 


8 


22 


.4040 


68 


53 


1.3270 


37 


83 


8.1443 


7 


23 


.4245 


67 


54 


1.3764 


36 


84 


9.5144 


6 


24 


.4452 


66 


55 


1.4281 


35 


85 


11.43 


5 


25 


.4663 


65 


56 


1.4826 


34 


86 


14.30 


4 


26 


.4877 


64 


57 


1.5399 


33 


87 


19.08 


3 


27 


.5095 


63 


58 


1.6003 


32 


88 


24.64 


2 


28 


.5317 


62 


59 


1.6643 


31 


89 


57.29 


1 


29 


.5543 


61 


60 


1.7321 


30 


90 


0.0000 





30 


.5774 


60 








■ 







FORMULAS AND TABLES 313 



LIST OF TABLES GIVEN IN THIS BOOK 

PAGE 

Curvature of earth's surface 28 

Cosines 311 

Cotangents 312 

Day, length of longest day at different latitudes 158 

Declination of the sun, see analemma 127 

Deviation of freely swinging pendulum due to earth's rotation . 57 

Distances, etc., of planets 266 

Equation of time, see analemma 127 

Earth's dimensions, etc 310 

Latitudes, lengths of degrees 44 

of principal cities of the world 88 

Longitudes, lengths of degrees 44 

of principal cities of the world 88 

Measures of length 310 

Meridional parts 217 

Sines, natural 311 

Solar systeni table 266 

Standard time adoptions 81 

Tangents, natural 312 

Time used in various countries 81 

Velocity of earth's rotation at different latitudes 58 

Vertical ray of sun, position on earth, see analemma 127 



GLOSSARY 

Aberration, the apparent displacement of sun, moon, planet, or star pro- 
duced as a resultant of (a) the orbital velocity of the earth, and (6) 

the velocity of light from the heavenly body. 
Acceleration, increase or excess of mean motion or velocity. 
Altitude, elevation in degrees (or angle of elevation) of an object above 

the horizon. 
Analemma, a scale showing (a) the mean equation of time and (6) 

the mean declination of the sun for each day of the year. 
Aphelion (S, fe' li on), the point in a planet's orbit which is farthest 

from the sun. 
Apogee (Sp' o je), the point farthest from the earth in any orbit; 

usually applied to the point in the moon's orbit farthest from the 

earth. 
Apparent solar day, see Day. 
Apparent (solar) time, see Time. 
Apsides (Sp' si dez), line of, a line connecting perihelion and aphelion 

of a planet's orbit, or perigee and apogee of a moon's orbit. 

Apsides is plural for apsis, which means the point in an orbit 

nearest to the primary or farthest from it. 
Arc, part of a circle; in geography, part of the circumference of a 

circle. 
Asteroids, very small planets. A large number of asteroids revolve 

around the sun between the orbits of Mars and Jupiter. 
Autumnal equinox, see Equinox. 
Axis, the line about which an object rotates. 
Azimuth (Sz' i muth) the angular distance of an object from the celestial 

meridian of the place of the observer to the celestial meridian of 

the object. The azimuth of the sun is the distance in degrees from 

its point of rising or setting to a south point on the horizon. 
Celestial sphere, the apparent hollow sphere in which the sun, moon, 

planets, comets, and stars seem to be located. 
Center of gravity the point about which a body (or group of 

bodies) balances. 
Centrifugal force (sen trif u gal), a force tending away from a center. 
Centripetal force (sen trip' e tal), a force tending toward a center. 
Colures (ko lurz'), the four principal meridians of the celestial sphere, 

two passing through the equinoxes and two through the solstices. 
Conjunction, see Syzygy. 

314 



GLOSSARY 316 

Copernican system (ko per' ni can), the theory of the solar system 

advanced by Copernicus (1473-1543) that the sun is the center of 

the solar system, the planets rotating on their axes and revolving 

around the sun. See Heliocentric theory. 

Cotidal lines, lines passing through places that have high tide at the 

same time. 
Day. 
AsTKONOMicAL DAY, a period equal to a mean solar day, reckoned 
from noon and divided into twenty-four hours, usually numbered 
from one to twenty-four. 
Civil day, the same as an astronomical day excepting that it is 
reckoned from midnight. It is also divided into twenty-four 
hours, usually numbered in two series, from one to twelve. 
SiDEREAi. DAY, the interval between two successive passages of a 
celestial meridian over a given terrestrial meridian. The zero 
meridian from which the sidereal day is reckoned is the one 
passing through the First point of Aries. The length of the 
sidereal day is 23 h. 56 m. 4.09 s. The sidereal day is divided 
into twenty-four hours, each shorter than those of the civil or 
astronomical day; they are numbered from one to twenty-four. 
Solar day 

Apparent solar day, the interval between two successive passages 
of the sun's center over the meridian of a place; that is, from 
sun noon to the next sun noon; this varies in length from 23 h. 
59 m. 38.8 s. to 24 h. m. 30 s. 

Mean solar day, the average interval between successive passages 
of the sun's center over the meridian of a place; that is, the aver- 
age of the lengths of all the solar days of the year; this average 
is 24 h. as we commonly reckon civil or clock time. 
Declination is the distance in degrees of a celestial body from the 
celestial equator. Declination in the celestial sphere corresponds 
to latitude on the earth. 
Eccentricity (6k s6n tris' I ty), see Ellipse. 

Ecliptic (S klip' tik), the path of the center of the sun in its apparent 
orbit in the celestial sphere. A great circle of the celestial sphere 
whose plane forms an angle of 23° 27' with the plane of the equator. 
This inclination of the plane of the ecliptic to the plane of the 
equator is called the obliquity of the ecliptic. The points 90° 
from the ecliptic are called the poles of the ecliptic. Celestial 
latitude is measured from the ecliptic. 
Ellipse, a plane figure bounded by a curved line, every point of which 
is at such distances from two points within called the foci (pro- 
nounced to' si; singular, focus) that the sum of the distances is 
constant. 



316 GLOSSARY 

Eccentricity (6k sgn tris' i ty) is the fraction obtained by dividing 
the distance of a focus to the center of the major axis by one half 
the major axis. 
Oblateness or eUipticity is the deviation of an ellipse from a circle 
and is the fraction obtained by dividing the difference between 
the major and minor axes by the major axis. 
EUipticity (61 lip tis' i ty), see Ellipse. 

Equation of time (e kwa' shun), the difference between apparent 
solar time, or time as actually indicated by the sun, and the mean 
solar time, or the average time indicated by the sun. It is usually 
indicated by the minus sign when the apparent sun is faster than 
the mean sun and with the plus sign when the apparent time is 
slow. The apparent sun time combined with the equation of time 
gives the mean time; e.g., by the apparent sun it is 10 h. 30 m., the 
equation is — 2 m. (sun fast 2 m.), combined we get 10 h. 28 m., 
the mean sun time. See Day. 
Equator (e kwa' ter), when not otherwise qualified means terrestrial 
equator. 

Celestial EQaATOR, the great circle of the celestial sphere in the 
plane of the earth's equator. Declination is measured from the 
celestial equator. 
Terrestrial equator, the great circle of the earth 90° from the 
poles or ends of the axis of rotation. Latitude is measured from 
the equator. 
Equinox, one of the two points where the ecliptic intersects the celes- 
tial equator. Also the time when the sun is at this point. 
Autumnal equinox, the equinox which the sun reaches in autumn. 

Also the time when the sun is at that point, September 23. 
Vernal equinox, the equinox which the sun reaches in spring. 
This point is called the First point of Aries, since that sign of 
the zodiac begins with this point, the sign extending eastward 
from it 30°. The celestial meridian (see colure) passing through 
this point is the zero meridian of the celestial sphere, from which 
celestial longitude is reckoned. The vernal equinox is also the 
time when the sun is at this point, about March 21, the beginning 
of the astronomical year. See Year. 
Geocentric (je 6 s6n' trik ; from ge, earth; centrum, center), 
Theory of the solar system assumes the earth to be at the center 

of the solar system ; see Ptolemaic system. 
Latitude, see Latitude. 
Parallax, see Parallax. 
Geodesy (je 5d' 8 sy), a branch of mathematics or surveying which 
is applied to the determination, measuring, and mapping of lines or 
areas on the surface of the earth. 



GLOSSARY 317 

Gravitation, the attractive force by which all particles of matter tend 

to approach one another. 
Gravity, the resultant of (a) the earth's attraction for any portion of 
matter rotating with the earth and (6) the centrifugal force due to 
its rotation. The latter fol-ce (6) is so small that it is usually 
ignored and we commonly speak of gravity as the earth's attrac- 
tion for an object. Gravity is still more accurately defined in the 
Appendix. 
Heliocentric (hg li o sgn'trik ; from helioB, sun ; centrum, center). 

Theory of the solar system assumes the sun to be at the center of 
the solar system; also called the Copernican system (see Coper- 
nican system). 
Parallax, see Parallax. 
Horizon (ho ri' zon), the great circle of the celestial sphere cut by a 
plane passing through the eye of the observer at right angles to 
the plumb line. 

Dip of horizon. If the eye is above the surface, the curvature of 
the earth makes it possible to see beyond the true horizon. The 
angle fonned, because of the curvature of the earth, between the 
true horizon and the visible horizon is called the dip of the 
horizon. 
Visible horizon, the place where the earth and sky seem to meet. 
At sea if the eye is near the surface of the water the true horizon 
and the visible horizon are the same, since water levels and forms 
a right angle to the plumb line. 
Hour-circles, great circles of the celestial sphere extending from pole 
to pole, so called because they are usually drawn every 15° or one 
for each of the twenty-four hours of the day. While hour-circles 
correspond to meridians on the earth, celestial longitude (see 
Longitude) is not reckoned from them as they change with the 
rotation of the earth. 
Latitude, when not otherwise qualified, geographical latitude is 
meant. 

Astronomical latitude, the distance in degrees between the 
plumb line at a given point on the earth and the plane of the 
equator. 
Celestial latitude, the distance in degrees between a celestial 

body and the ecliptic. 
Geocentric latitude, the angle formed by a line from a given 
point on the earth to the center of the earth (nearly the same 
as the plumb line) and the plane of the equator. 
Geographical latitude, the distance in degrees of a given point 
on the earth from the equator. Astronomical, geocentric, and 



318 GLOSSARY 

geographical latitude are nearly the same (see discussion of Lati- 
tude in Appendix). 
Local time, see Time. 
Longitude. 
Celestial longitude, the distance in degrees of a celestial body 
from lines passing through the poles of the ecliptic (see Ecliptic), 
called ecliptic meridians; the zero meridian, from which celestial 
longitude is reckoned, is the one passing through the First point 
of Aries (see Equinox). 
Terrestrial longitude, the distance in degrees of a point on the 
earth from some meridian, called the prime meridian. 
Mass, the amount of matter in a body, regardless of its volume or 

size. 
Mean solar time, see Time. 
Meridian. 
Celestial meridian, a great circle of the celestial sphere passing 
through the celestial poles and the zenith of the observer. The 
celestial meridian passing through the zenith of a given place 
constantly changes with the rotation of the earth. 
Terrestrial meridian, an imaginary line on the earth passing 
from pole to pole. A meridian circle is a great circle passing 
through the poles. 
Month. 

Calendar month, the time elapsing from a given day of one month 
to the same numbered day of the next month; e.g., January 3 
to February 3. This is the civil or legal month. 
Sidereal month, the time it takes the moon to revolve about the 
earth in relation to the stars; one exact revolution of the moon 
about the earth; it varies about three hours in length but aver- 
ages 27.32166 d. 
Synodic month, the time between two successive new moons or 
full moons. This is what is commonly meant by the lunar 
month, reckoned from new moon to new moon; its length varies 
about thirteen hours but averages 29.53059 d. There are sev- 
eral other kinds of lunar months important in astronomical 
calculations. 
Solar month, the time occupied by the sun in passing through a 
sign of the zodiac; mean length, 30.4368 d. 
Hadir (na' der), the point of the celestial sphere directly under the 

place on which one stands; the point 180° from the zenith. 
Neap tides, see Tides. 

Nutation, a small periodic elliptical motion of the earth's axis, due 
principally to the fact that the plane of the moon's orbit is not the 
same as the plane of the ecliptic, so that when the moon is on one 



GLOSSARY 319 

side of the plane of the ecliptic there is a tilting tendency given the 
bulging equatorial region. The inclination of the earth's axis, or 
the obUquity of the ecliptic, is thus slightly changed through a 
period of 18.6 years, varying each year from 0" to 9.2". (See 
Motions of the Axis in the Appendix.) 
Oblateness, the same as elliptieity; see Ellipse. 
Oblate spheroid, see Spheroid. 

Obliquity (ob lik' wi ty), of the ecliptic, see Ecliptic. 
Opposition, see Syzygy. 
Orbit, the path described by a heavenly body in its revolution about 

another heavenly body. 
Parallax, the apparent displacement, or difference of position, of an 
object as seen from two different stations or points of view. 
Annual or heliocentric parallax of a star is the difference in 
the star's direction as seen from the earth and from the sun. 
The base of the triangle thus formed is based upon half the major 
axis of the earth's orbit. 
Diurnal or geocentric parallax of the sun, moon, or a planet is 
the difference in its direction as seen from the observers' station 
and the center of the earth. The base of the triangle thus formed 
is half the diameter of the equator. 
Perigee (p6r' i je), the point in the orbit of the moon which is nearest 
to the earth. The term is sometimes applied to the nearest point 
of a planet's orbit. 
Perihelion (p5r i hS' li on), the point in a planet's orbit which is nearest 

to the sun. 
Poles. 
Celestial, the two points of the celestial sphere which coincide 
with the earth's axis produced, and about which the celestial 
sphere appears to rotate. 
Or THE ecliptic, the two points of the celestial sphere which are 90° 

from the ecliptic. 
Terrestrial, the ends of the earth's axis. 
Ptolemaic system (tol e ma' ik), the theory of the solar system advanced 
by Claudius Ptolemy (100-170 a.d.) that the earth is the center of 
the universe, the heavenly bodies daily circhng around it at different 
rates. Called also the geocentric theory (see Geocentric). 
Radius (plural, radii, ra' di i), half of a diameter. 

Radius Vector, a line from the focus of an ellipse to a point in the 

boundary line. Thus a line from the sun to any planet is a radius 

vector of the planet's orbit. 

Refraction of light, in general, the change in direction of a ray of 

light when it enters obliquely a medium of different density. As 



320 GLOSSARY 

used in astronomy and in this work, refraction is the change in 
direction of a ray of light from a celestial body as it enters the 
atmosphere and passes to the eye of the observer. The eif eot is to 
cause it to seem higher than it really is, the amount varying with the 
altitude, being zero at the zenith and about 36' at the horizon. 
Revolution, the motion of a planet in its orbit about the sun, or of a 

satellite about its planet. 
Rotation, the motion of a body on its axis. 
Satellite, a moon. 
Sidereal day, see Day. 
Sidereal year, see Year. 
Sidereal month, see Month. 
Sidereal time, see Time. 
Signs of the zodiac, its division of 30° each, beginning with the vernal 

equinox or First point of Aries. 
Solar times, see Time. 

Solstices (sol' stis es; sol, sun; stare, to stand), the points in the 
ecliptic farthest from the celestial equator, also the dates when the 
sun is at these points; June 21, the summer solstice; December 22, 
th« winter solstice. 
Spheroid (sfe' roid), a body nearly spherical in form, usually referring 
to the mathematical form produced by rotating an ellipse about 
one of its axes; called also an ellipsoid or spheroid of revolution 
(in this book, a spheroid of rotation). 
Oblate spheroid, a mathematical solid produced by rotating an 

ellipse on its minor axis (see Ellipse). 
Prolate spheroid, a mathematical solid produced by rotating an 
ellipse on its minor axis (see Ellipse). 
Syzygy (siz' i jy; plural, syzygies), the point of the orbit of the moon 
(planet or comet) nearest to the earth or farthest from it. When 
in the syzygy nearest the earth, the moon (planet or comet) is said 
to be in conjunction ; when in the syzygy farthest from the earth it 
is said to be in opposition. 
Time. 
Apparent solar time, the time according to the actual position 
of the sun, so that twelve o'clock is the moment when the 
sun's center passes the meridian of the place (see Day, apparent 
solar). 
Astronomical time, the mean solar time reckoned by hours num- 
bered up to twenty-four, beginning with mean solar noon (see 
Day, astronomical). 
Civil time, legally accepted time ; usually the same as astronomical 
time except that it is reckoned from midnight. It is commonly 
numbered in two series of twelve hours each day, from midnight 



GLOSSARY $21 

and from noon, and is based upon a meridian prescribed by law 
or accepted as legal (see Day, civil). 

Equation of time, see Equation of time. 

SiDEKEAL TIME, the time as determined from the apparent rotation 
of the celestial sphere and reckoned from the passage of the 
vernal equinox over a given place. It is reckoned in sidereal 
days (see Day, sidereal). 

Solar time is either apparent solar time or mean solar time, 
reckoned from the mean or average position of the sun (see 
Day, solar day). 

Standard time, the civil time that is adopted, either by law or 
usage, in any given region; thus practically all of the people of 
the United States use time which is five, six, seven, or eight hours 
earlier than mean Greenwich time, being based upon the mean 
solar time of 75°, 90°, 105°, or 120° west of Greenwich. 
Tropical year, see Year. 
Tropics. 

Astronomical, the two small circles of the celestial sphere parallel 
to the celestial equator and 23° 27' from it, marking the north- 
ward and southward limits of the sun's center in its annual 
(apparent) journey in the ecliptic; the northern one is called the 
tropic of Cancer and the southern one the tropic of Capricorn, 
from the signs of the zodiac in which the sun is when it reaches 
the tropics. 

Geographical, the two parallels corresponding to the astronomical 
tropics, and called by the same names. 
Vernal equinox, see Equinox, vernal. 
Year. 

Anomalistic tear (a nom a lis' tik), the time of the earth's revo- 
lution from perihelion to perihelion again; length 365 d., 6 h., 
13 m., 48 s. 

Civil tear, the year adopted by law, reckoned by all Christian 
countries to begin January 1st. The civil year adopted by 
Protestants and Roman Catholics is almost exactly the true 
length of the tropical year, 365 . 2422 d., and that adopted by Greek 
Catholics is 365 . 25 d. The civil year of non-Christian countries 
varies as to time of beginning and length, thus the Turkish civil 
year has 354 d. 

Lunar tear, the period of twelve lunar sjmodioal months (twelve 
new moons); length, 354 d. 

Sidereal tear, the time of the earth's revolution around the sun 
in relation to a star; one exact revolution about the sun; length, 
365.2564 d. 

JO. mats. GEO. — 21 



322 GLOSSAEY 

Tropical year, the period occupied by the sun in passing from one 

tropic or one equinox to the same again, having a mean length of 

365 d. 5 h. 48 m. 45.51 s. or 365.2422 d. A tropical year is shorter 

than a sidereal year because of the precession of the equinoxes. 

Zenith (zS' nith), the point of the celestial sphere directly overhead; 

180° from the nadir. 
Zodiac (za' di ak), an imaginary belt of the celestial sphere extending 
: about eight degrees on each side of the ecliptic. It is divided into 
twelve equal parts (30° each) .called signs, each sign being somewhat 
to the west of a constellation of the same name. The ecUptic being 
the central line of the zodiac, the sun is always in the center of it, 
apparently traveling eastward through it, about a month in each 
sign. The moon being only about 5° from the ecliptic is always in 
the zodiac, traveling eastward through its signs about 13° a day. 



INDEX 



Abbe, Cleveland, 5, 65. 

Abbott, Lyman, 76, 142-144. 

Aberdeen, S. D., 89. 

Aberration of light (see Glos- 
sary, p. 314), 104^106, 110, 
278. 

Abraham, 141. 

Acapulco, Mexico, 101. 

Acceleration (see Glossary, p. 
314), 290-292. 

Adelaide, Australia, 88. 

Aden (a'den), Arabia, 88. 

Africa, 186, 200, 214, 216, 223, 
268. 

Akron, O., 81. 

Alabama, 90. 

Alaska, 69, 87, 89, 91, 98, 102, 
103. 

Albany, N. Y., 89. 

Albany, Tex., 243. 

Alberta, Canada, 82. 

Aleutian Is., 96. 

Alexandria, Egypt, 32, 83, 88, 
270, 271. 

Algeria, 83. 

Allegheny Observatory, Alle- 
gheny, Pa., 69. 

AUen, W. F., 66. 

Allowance for curvature of 
earth's surface, 226, 230, 
233. 

Al-Mamoum, 271. 

Almanac, 118, 123-125, 171. 

Atitude, of noon sun, 12, 13, 
170-174, 302, 303. 
of polestar or celestial pole, 
58-61, 170-174. 



Amazon, bores of, 189. 
American Practical Navigator, 

217. 
Amsterdam, Holland, 82, 88. 
Analemma (see Glossary, p. 

314), description of, 126. 
representation of, 127. 
uses of, 128-130, 171-174, 

301. 
Anaximander (&n a,x i m&n'der), 

269. 
Annapolis, Md., 89. 
Ann Arbor, Mich., 89. 
Antipodal (&n tip'6 dal) areas, 

map showing, 41. 
Antwerp, Belgium, 88. 
Apheliom (see Glossary, p. 314), 

119, 285, 287. 
Apia (a pe'a), Samoa, 88. 
Apogee (see Glossary, p. 314) 

178. 
Apsides (see Glossary, p. 314) 

287. 
Aquarius (a kwa'rl us), 296. 
Arabia, 88. 

Arcturus (ark tii'rus), 54, 109. 
Area method of determining 

geoid, 36, 37. 
Area of earth's surface, 310. 
Arequipa (a ra ke'pa), Peru, 

164. 
Argentinajar j§n te'na), 81. 
Aries (a'ri ez) , constellation, 294. 
First point of, 294, 316. 
sign of zodiac, 118, 297, 298. 
Aristarchus (fi,r is tar'kiis), 275, 

277. 
Aristotle (&r'is tot 1) , 270. 
Arkansas (ar'kftnsa ), 90, 233. 



323 



324 



INDEX 



Arkansas River, 233. 
Armenian Churcli, 145. 
Asia, 34, 41, 186, 214. 
Asteroids, 50, 246, 253, 293, 314. 
Astronomical day, 130, 315. 
Athens, Greece, 42, 84, 88. 
Atlanta, Ga., 89. » 
Atlantic Ocean, 141, 186, 242. 
Atmosphere, 161-K57. 

absence of, on moon, 263, 264. 

how heated, 167. 

on Jupiter, 257. 

on Mars, 254. 

on Mercury, 261. 

on Venus, 256. 

origin of, 251, 252. 
Attu Island, 89, 99. 
Auckland, New Zealand, 97., 
Augusta, Me., 89. 
Augustan calendar, 135. 
Austin, Tex., 89. 
AustraUa, 41, 96, 97, 101. 
Austria-Hungary, 68, 82. 
Axis, changes in position of, 288, 
289. 

defined, 22, 314. 

inclination of, see ObHquity of 
echptic. 

parallelism of, 154. 
Azimuth, 303, 314. 

B 

Babinet, 207. 

Bacchus, 135. 

Bailey, S. I., 163. 

Balearic Is., 86. 

Balkan States, 140, 145. 

Ball's History of Mathematics, 
269. 

Baltimore, Md., 89. 

Bangor, Me., 89. 

Bankok, Siam, 88. 

Barcelona, Spain, 88. 

Barlow and Bryan's Mathe- 
matical Astronomy, 292. 

Barometer, 161, 176. 



Base line, 231-234, 243, 244. 

Batavia, Java, 88. 

Behring Strait, 33. 

Belgium, 68, 82, 88. 

Beloit, Wis., 89. 

Benetnasoh, 61. 

Bergen, Norway, 88. 

Berkeley, CaHf., 89. 

Berlin, Germany, 88. 

Bessel, F. W., 31, 32, 35, 109. 

Bethlehem, 145. 

Big Dipper, 9,48, 60, 61, 111, 295. 

Bismarck Archipelago, 83. 

Bismarck, N. D., 89. 

Black HiUs Meridian, 230. 

Bogota (bo go ta') , Columbia, 83. 

Boise (boi'za), Ida., 89. 

Bombay (b6m ba'), India, 88. 

Bonne's projection, 221, 224. 

Bonn Observatory, 109. 

Bordeaux (b6rd6'), France, 88. 

Bores, tidal, 187, 189. 

Bosphorus, 144. 

Boston, Mass., 67, 89. 

Bowditch, Nathaniel, 217. 

Brahe (bra), Tycho, 109, 277, 
278. 

Bradley, James, 106, 278. 

Brazil, 89. 

British Columbia, 82. 

British Empire, 82. 

Brussels, Belgium, 82, 88. 

Budapest (b66' da pest), Hun- 
gary, 86. 

Buenos Aires (bo'niis a'riz), 
Argentina, 73, 88. 

Buffalo, N. Y., 90. 

Bulgaria, 68. 

Bulletin, U. S. G. S., 243. 

Burmah, 82. 



Cadiz (ka'dJz), Spain, 88. 
Caesar, Augustus, 135, 136. 

Julius, 135, 138. 
Cairo (ki'ro), Egypt, 83, 88. 



INDEX 



325 



Calais, Me., 32. 
Calcutta, India, 88, 108. 
Calendar, 132-145. 

ancient Mexican, 141. 

Augustan, 135. 

Chaldean, 141. 

Chinese, 141. 

early Roman, 134, 138. 

Gregorian, 136. 

Jewish, 144, 190. 

Julian, 135, 140. 

Mohammedan, 140, 142-144. 

on moon, 262. 

Turkish, 142. 
Cahfornia, 32, 35, 37, 68, 89, 90, 

91, 155, 168. 
Callao (kel la'6), Peru, 86. 
Canada, 48, 229, 233. 
Cancer, constellation of, 150. 

sign of zodiac, 295. 

tropic of, 150, 269. 
Canterbury Tales, quoted, 118. 
Canton, China, 88. 
Cape Colony, Africa, 82. 
Cape Deshnef, Siberia, 96, 99. 
Cape Horn, 41. 
Cape May, N. J., 32, 35. 
Cape Town, Africa, 32, 73. 
Capricorn, constellation of, 150. 

sign of zodiac, 296, 298. 

tropic of, 151. 
Caracas (ka ra'kas), Venezuela, 

87. 
Carleton College, Northfield, 

Minn., 69. 
Carnegie Institution of Wash- 
ington, 35. 
Carolines, the,_83. 
Cassini (kas se'ne), G. D., and 

J., 274. 
Cassiopeia (k&s si o pe'ya), 9, 60, 

61. 
Cayenne (kl gn'), French Guiana, 

28, 29, 88. 
Celestial, equator, 47, 150, 170- 
174, 283, 284, 294, 316. 



Celestial latitude, 283, 284. 
longitude,' 284, 318. 
meridians, 283, 284. 
pole, 46, 47, 68, 61, 170-174. 
sphere, 45-51, 62, 314. 
tropics, 150, 151, 321. 
Central time, in Europe, 68, 76, 

85, 86. 
in the United States, 67, 71, 

74, 75, 77, 128. 
Centrifugal force (see Glossary, 

p. 314), 13-16, 29, 51, 279- 

283, 290, 309. 
Centripetal force (see Glossary, 

p. 314), 15, 16. 
Ceres, 135. 
Ceylon (se 16n'), 82. 
Chaldeans, 134, 141, 269, 294, 

295. 
Chamberlin, T. C, 5, 250. 
Charles IX., King of France, 

139. 
Charleston, S. C, 90. 
Charles V., Emperor of Spain, 

214. 
Chatham Islands, 82, 96. 
Chaucer, quoted, 118. 
Cheyenne (shi 6n'), Wyo., 90. 
Chicago, lU., 63, 67, 90, 95, 96, 

98, 129, 174, 208. 
Chile (che'la), 82, 89. 
China, 83, 88, 89, 100, 163, 269. 
Chinese, calendar, 141. 

zodiac, 296. 
Choson, Land of the Morning 

Calm (Korea)^, 97. 
Christiania (kris te a'ne a), Nor- 
way, 88. 
Christmases, three in one year, 

145. 
Chronograph (krSn'o graph), 69. 
Chronometer (kron6m'e ter), 64. 
Cincinnati, 0., 74, 90. 
Circumference of earth, 30, 31, 

310. 
Circle defined, 20, 21. 



326 



INDEX 



Circle of illumination, or day 

circle, 149, 151, 152, 155-157. 
Civil day, 130, 315. 
Clarke, A. R., 30-32, 34, 35, 37, 

44, 275, 310. 
Cleomedes (kle 6m'e dez), 272. 
Cleveland, 0., 74, 90. 
Clock, sidereal, 69, 70. 
Collins, Henry, lOO! 
Colon (ko Ion'), Panama, 85. 
Colorado, 67, 90. 
Color vibrations, 106, 107. 
Columbia, 83. 
Columbia, S. C, 90. 
Columbus, Christopher, 137, 138, 

212, 272. 
Columbus, O., 74, 90. 
Comets, 50, 246, 278. 
Compass, magnetic, or mariner's, 

152, 153, 226, 227. 
Concord, N. H., 90. 
Congressional township, 228. 
Conic projection, 218-224. 
Conjunction, 178, 185, 320. 
Connecticut, 90. 
Constantinople, Turkey, 88. 
Convergence of meridians, 230, 

233. 
Copenhagen, Denmark, 88. 
Copernican system (see Glos- 
sary, p. 315), 276-278. 
Copernicus (ko per'nl kus), 50, 

109, 276-278. 
Cordoba (kor'do ba), Argentina, 

81. 
Corinto (ko ren'to), Nicaragua, 

85. 
Correction line, 234. 
Cosines, natural, table of, 311. 
Costa Rica, 83. 
Cotangents, natural, table of, 

312 
Cotidal iines, 186, 315. 
Crepusculum, the, 165. 
Creston, Iowa, 80. 
Cuba, 69, 83, 88. 



Cuidad Juarez, see Juarez. 

Curvature of surface of earth, 
rate of, 27, 28, 43, 44, 226, 
233. 

Cygnus (sig'ntts ; plural and pos- 
sessive singular, cygni), 109. 

Cylindrical projection, 209-218, 
224. 

D 

Dakotas, division of, 234. 

Danish West Indies, 83. 

Date line, see International 

date line. 
Day (see Glossary, p. 315), as- 
tronomical, 130. 
circle, see Circle of illumina- 
tion, 
civil, 130. 
length of, 155-158. 
lunar, 188. 

origin of names of days of 
week, 142. 
> sidereal, 114, 130. 
solar, 62, 114, 130. 
total duration of a, 98. 
Deadwood, S. D., 90. 
Deimos (di'mus), 255. 
Declination (see Glossary, p. 
315), 125, 127, 171-175, 
284. 
De la Hire, Philhppe (fe lep' de 

laer'),201. 
Denmark, 68, 83, 88, 137. 
Density, formula for, 309. 
Density of earth, 310. 
Denver, Col., 26, 27, 67, 90. 
Des Moines (de moin'), Iowa, 90. 
Deshnef, Cape, 96, 99. 
Detroit, Mich., 63, 73, 90. 
Deviation, of pendulum, 54-57. 

of plumb Hne, 281-283. 
Dewey, George, 103. 
Diameter of earth, 29-31, 45, 

310. 
Dimensions of earth, 310. 



INDEX 



327 



Dip off horizon (see Glossary, p. 

317). 
Distances, of planets, 266, 310. 

of stars, 45, 246. 
District of Columbia, 77, 91, 124. 
Diurnal (diur'nal), motion of 

earth, see Rotation. 
Division of Dakotas, 234. 
Dryer, Charles R., 5. 
Dublin, Ireland, 64, 82, 88. 
Duluth, Minn., 90. 

E 

Earth in Space, 246-267. 

Earth's dimensions, 310. 

Eastern time, in Europe, 68, 87. 
in the United States, 66, 67; 
■ 71, 75, 78. 

Eastward deflection of falling 
obj;cts, 51-54. 

Eclipse, 24, 116, 161, 178. 

Ecliptic (see Glossary, p. 315), 
116, 119, 284, 286, 287. 
obliquity of, 118, 147, 288, 
310. 

Edinburgh (6d'in bxir ro), Scot- 
land, 88. 

Egypt, 83, 88, 134, 268. 

El Castillo (61 kas tel'yo), Nica- 
ragua, 85. 

Ellipse (see Glossary, p. 315), 
20-22, 193, 206, 308. 

Ellipsoid of rotation (see Glos- 
sary, p.. 320), 36. 

El Ocotal (gl ok o tal'), Nica- 
ragua, 85. 

El Paso, Tex., 68, 75, 76. 

Encyclopaedia Britannica, 101. 

England, 9, 80, 82, 89, 98, 101, 
137, 139, 274. 

Ephemeris (e i^m'e ris), see 
Nautical almanac. 

Epicureans, 272. 

Equador, 83. 

Equation of time (see Glossary, 
p, 316); 123-127. 



Equator (see Glossary, p. 316), 

celestial, 47, 150, 170-174, 

283, 284, 294. 
length of day at, 157. 
terrestrial, 23, 33, 48, 49, 118, 

148-152, 272, 280, 310. 
Equinox ( ee Glossary, p. 316), 

119, 147, 148, 155, 156, 168, 

169, 285. 
precession of, 286-288, 303. 
Eratosthenes (er a t6s' the nez), 

270, 271. 
Erie, Pa., 90. 
Establishment, the, of a port, 

181. 
Eudoxus, 270. 
feuripides (tirip'Idez), 122. 
Europe, 101, 166, 168, 220, 222, 

223, 226, 288, 294. 



Fargo, N. D., 90. 

Farland, R. W., 132. 

Faroe (fa'ro). Islands, 83. 

Fathom, length of, 310. 

Fiji Islands, 96, 100. 

Fiske, John, 145, 268. 

Fixed stars, 10, 108, 109, 265, 

266. 
Florence, Italy, 88. 
Florida, 90, 91. 
Form of the earth, 24-44. 
Formosa, 84. 
Formulas, 307-309. 
Foucault (f66 ko'), experiment, 

with gyroscope, 155. 
with pendulum, 54r-57. 
France, 32, 64, 83, 88, 89, 130, 

137, 139, 187. 
Franklin's almanac, 137. 
Fundy, Bay of, 189. 

G 

Gainesville, Ga., 75. 
Galilei, Galileo (gal I le'6 
gai i la'e), 277, 278. 



328 



INDEX 



Galveston, Tex., 75, 90. 

Gannett, Garrison and Hous- 
ton's Commercial Geography, 
206. 

Gauss, 52. 

Gemini, 295, 298. 

Genesis, 141. 

Genoa, Italy, 76. 

Geocentric, latitude, see Lati- 
tude, 
theory (see Glossary, p. 316), 
277, 278. 

Geodesy (see Glossary, p. 316), 
275. 

Geodetic Association, Interna- 
tional, 289. 

George II., King of England, 
227. 

Geographical constants, 310. 

Geoid (je'oid), 33-37, 275. 

Geometry, origin of, 226. 

Georgia, 68, 75, 80, 89, 91. 

German East Africa, 84. 

Germany, 68, 77, 83, 84, 88, 89, 
97, 137, 233. 

Gibraltar, Spain, 82, 88. 

Glasgow, Scotland, 88. 

Globular projection, 198-201, 
211. 

Glossary, 314-322. 

Gnomonic (no mdn'tlc) , cylin- 
drical projection, 209-211. 

Gnomonic projection, 201-204, 
211. 

Goode, J. Paul, 302. 

GoodseU Observatory, North- 
field, Minn., 69. 

Gravimetric lines, map show- 
ing, 34. 

Gravitation, 16-18, 178, 179, 
272 317. 

Gravity' 18,' 25, 28, 29, 183- 
185, 279-282, 290, 304, 305, 
308, 309, 317. 
on Jupiter, 19, 256. 
on Mars, 253. 



Gravity on Mercury, 260. 
on moon, 19, 20, 261. 
on Neptune, 259. 
on Saturn, 258. 
on sun, 19, 264, 265. 
on Uranus, 259. 
on Venus, 256. 
Great Britain, 64, 68, 77, 80, 

182 233 
Great circle saihng, 203, 204, 

212. 
Greece, 84, 88. 
Greenland, 218. 

Greenwich (Am. pron., grfen' 
wich; Eng. pron., grin'ij oi 
grgn'ij), England, 41, 42, 
64, 67, 68, 73, 77, 78, 80, 
82, 84, 86-91, 95, 100, 124- 
126, 172. 
Gregorian calendar, 136. 
Guam, 73, 87. 
Guaymas, Mexico, 85. 
Guiana, French, 88, 273. 
Gulf of Mexico, 37, 189. 
Gunnison, Utah, 243. 
Guthrie, Okla, 90. 
Gyroscope (ji'ro skop), 154, 155. 

H 

Hague, The, Holland, 84, 88. 
Hamburg, Germany, 88, 125. 
Harkness, William, 275, 310. 
Harper's Weekly, 1^3, 164. 
Harte, Bret, 92. 
Hartford, Conn., 90. 
Harvard Astronomical Station 

(Peru), 164. 
Havana, Cuba, 83, 87, 88. 
Hawaiian (Sandwich) Islands, 

87, 100. 
Hayden, E. E., 5, 73, 77. 
Hayford, J. F., 36, 37, 245. 
Hegira, 140. 

Heliocentric theory (see Glos- 
I sary, p. 317), 277, 278. 



INDEX 



329 



Heliodon (he' H o don), 303. 
Helena, Mont., 90. 
Hemispheres xmequally heated, 

169, 284. 

Heraclitus (hgr akli'tus), 270. 

Hercules (her'cti lez), constella- 
tion, 247. 

Herodotus (he r6d' o tus), 132, 
268. 

Herschel, John, 34. 

Hidalgo, Mexico, 85. 

Hipparchus (hip ar'kus), 271, 
286. 

Historical sketch, 268-278. 

Holland, 68, 84, 88, 89, 137. 

Holway, R. S., 155. 

Homer, 268. 

Homolographic projection, 205- 
207, 211. 

Honduras, 84. 

Hongkong, 82, 83, 88, 101. 

Honolulu, Hawaiian Islands, 
90, 100. 

Horizon (see Glossary, p. 317), 
38-40, 47, 152, 153, 158,' 

170, 175. 
Hungary, 137. 

Hutchins, Thomas, 228, 229. 
Huygens (hi'gens). Christian, 

273, 274. 



Iceland, 83. 
Idaho, 89. 

Illinois, 90, 230, 231. 
Impressions of a Careless Trav- 
eler, quoted, 76, 142-144. 
India, 32, 82, 88, 89, 186, 242. 
Indian principal meridian, 230. 
Indiana, 90, 230. 
Indianapolis, Ind., 90. 
Indian Ocean, 186. 
Indian Territory, survey of, 245. 
Insolation, 165-169. 
International date line, 95, 101. 



International Geodetic Associa- 
tion, 289. 

Intersecting conic projection, 
221, 222. 

Iowa, 80, 90. 

Ireland, 33, 64, 77, 82, 88, 233. 

Isle of Man, 82. 

Isogonal (i s6g'on al) line, 227. 

Italy, 68, 84, 88, 89. 



Jackson, Miss., 90. 
Jacksonville, Fla., 90. 
James II., King of England, 274. 
Japan, 84, 85, 89, 289. 
Java, 88. 

Jefferson, Thomas, 229. 
Jerusalem, 89. 
Journal of Geography, 155. 
Juarez (h66 a'reth), Mexico, 76. 
Julian calendar, 102. 
Jupiter, 19, 250, 253, 256, 257, 
266, 274, 277. 

K 

Kamerun, Africa, 84. 
Kansas, 35. 

Kansas City, Mo., 90, 96. 
Keewatin, Canada, 82. 
Kentucky, 79, 90. 
Kepler, Johann, 278. 

laws of, 284-286. 
Key West, Fla., 90. 
Kiaochau (ke a o chow'), China, 

83. 
Korea (ko re'a), 85. 
Kramer, Gerhard, 214. 
Kiistner, Professor, 109. 



La Condamine (lakon'damen), 

189. 
Lake of the Woods, 233. 
Lake Superior, 231. 
Landmarks, use of, in surveys, 

226-228. 



330 



INDEX 



Lansing, Mich., 90, 231. 
Lapland, 274. 
Larkin, E. L., 265-267. 
Latitude (see Glossary, p. 317), 

astronomical, 282. 

celestial, 283, 284. 

geocentric, 282. 

geographical, 42. 

determined by altitude of 

circumpolar star, 58-61. 
determined by Foucault 

experiment, 65, 56. 
determined by altitude of 
noon sun, 170-175. 

lengths of degrees, 42-44. 

of principal cities, 88-91. 

origin of term, 40. 
Law Notes, quoted, 81. 
Layard, E. L., 100. 
Leavenworth, Francis P., 5. 
Leo, 295, 298. 
Legal aspect of standard time, 

76-81. 
Leipzig, Germany, 89. 
Length of day, 155-158. 
Lewis, Ernest I., 145. 
Lexington, Ky., 90. 
Leyden, Holland, 84, 272. 
Libra (li'bra), 117, 295, 296. 
Lick Observatory, 69, 73. 
Lima (le'ma), Peru, 86. 
Lincoln, Keb., 90. 
Link of surveyor's chain, 310. 
Lisbon, Portugal, 41, 73, 86, 89. 
Little Dipper, 9. 
Little Rock, Ark., 90, 233. 
Liverpool, England, 89. 
London, England, 41, 63, 64, 

93-96, 99, 137. 
London Times, 139. 
Longitude (see Glossary, p. 318), 
and time, 62-91. 

celestial, 284. 

how determined, 63-65, 128. 

lengths of degrees, 44. 

of principal cities, 88-91. 



Longitude, origin of term, 40. 
Los Angeles, Calif., 90. 
Louisiana, 90, 230. 
Louisville, Ky., 79, 90. 
Louis XIV., King of France, 28. 
Lowell, Mass., 90. 
Lowell, Percival, 97, 254. 
Luxemburg, 68, 85. 
Luzon, 90. 

M 

Macaulay's History of England, 

138, 139. 
McNair, F. W.,5, 52. 
Madison, Wis., 67, 90. 
Madras (madras'), India, 73, 

82, 89. 
Madrid, Spain, 73. 
Magnetic compass, 152, 153, 226, 

227. 
Magnetic pole, 152, 153. 
Magellan's fleet, 92. 
Maine, 32, 37. 
Malta, 82. 

Managua (ma na'gua), Nica- 
ragua, 85. 
Manitoba, Canada, 82. 
Manila, Phihppine Is., 73, 90, 

101-103. 
change of date at, 101. 
Map, 41, 42, 230, 236. 
Map projections, 190-225. 
Mare Island Naval Observatory, 

69, 71, 87. 
Mariane Islands, 83. 
Markham, A. H., 153, 164, 165. 
Mars, 253-255, 266, 285, 306. 
Marseilles, France, 89. 
Maryland, 89. 
Massachusetts, 89, 90, 227. 
Mauritius (ma rish' i us) Island, 

73. 
Mean solar day (see Day). 
Measuring, diameter of moon, 

240, 241. 



INDEX 



331 



Measuring distances of objects, 
237, 241. 

heights of objects, 238. 
Measures of length, 310. 
Mediterranean, 101. 
Melbourne, Australia, 89. 
Memphis, Tenn., 90. 
Mercator projection, 204, 211- 

219, 221. 
Mercedonius, 135. 
Mercury, 183, 253, 260, 261, 266. 
Meridian, 23, 29, 32, 37, 38, 95- 
100, 187, 188, 190-225, 283, 
318. 

celestial, 283, 284, 318. 

circle, 23. 

length of degrees of, 44. 

prime, 41, 42. 

principal, for surveys, 230- 
236. 

rate for convergence, 233. 

standard time, 06-68, 71, 75, 
77, 78, 81-87. 302. 
Meridional parts, table of, 217. 
Meteors, 50, 248, 249. 
Meter, length of, 310. 
Metes and bounds, 226, 228. 
Mexico, 85,*89, 101. 
■ Gulf of, 189. 

Michigan, 51, 68, 74, 76, 89, 90. 
231. 

College of Mines, 52. 
Midnight sun, 163. 
Mile, in various coimtries, 310. 
Milwaukee, Wis., 90, 103. 
Mining and Scientific Press, 

52-54. 
Minneapolis, Minn., 63, 90, 96. 
Minnesota, 90, 91, 231, 233. 
Mississippi, 90. 

River, 231. 
Missouri, 90, 91. 

River, 227. 
Mitchell, Frank E., 5. 
Mitchell, S^D., 90. 
Miyako (me ya'ko) Islands, 84. 



Mobile, Ala., 90. 

Mohammedan calendar, 140, 142- 

144. 
MoUendo, Peru, 86. 
MoUweide projection, 207. 
Montana, 90. 

Montevideo, Uruguay, 87, 89. 
Montgomery, Ala., 90. 
Month (see Glossary, p. 318) 

133, 134. 
sidereal, 177, 188. 
synodic, 177. 
Moon or satelhte, 10, 11, 19, 161, 

176-185, 240, 241, 246, 255, 

257-259, 261-264, 266, 278, 

288, 297, 298, 308. 
Moore, G. B. T., 96. 
Morse, J. P., 303. 
Moscow, Russia, 89. 
Motion in the line of sight, 106- 

109. 
Motions of the earth, 289. 
Motions of the earth's axis, 286- 

280. 
Moulton, F. R., 250. 
Mountain time belt, 67, 68, 75. 
Mount Diablo meridian, 230. 
Munich (mu'nik), Germany, 89. 
Myths and superstitions of the 

zodiac, 297, 298. 

N 

Nadir (see Glossary, p. 318), 38. 

Naples, Italy, 89. 

Nash, George W., 4. 

Nashville, Tenn., 90. 

Natal, Africa, 82. 

Nautical almanac, 118, 124, 171. 

Neap tides, 185, 188. 

Nebraska, 81, 90, 91, 227. 

Nebulae, 248, 251. 

Nebular hypothesis, 247-252. 

Nehemiah, 141. 

Neptune, 253, 259, 260, 266. 

Neuchatel, Switzerland, 86. 



332 



INDEX 



Nevada, 91. 
^ewark, N. J., 90, 91. 
^ew Brunswick, Canada, 82. 
New Caledonia, 100. 
Newcomb, Simon, 118. 
Newchwang, China, 83. 
Newfoundland, 82. 
New Guinea, 83. 
New Hampshire, 90. 
New Haven, Conn., 90. 
New Jersey, 32, 78, 90, 91. 
New Mexico, 68, 76, 91, 230. 
New Orleans, La., 59, 67, 90, 108. 
New South Wales, 82, 89. 
New Style, 137-140, 143. 
Newton, Isaac, 15, 51, 138, 272. 
New York, 76, 78, 89-91, 95, 

98, 99, 204. 
New York Sun, 55. 
New Zealand, 82, 96, 97. 
Nicaea, Council of, 136. 
Nicaragua, 85. 
Nile, 226. 
North America, 100, 185, 213, 

219, 242, 287. 
North Carolina, 91. 
North, line, 11, 61, 130. 
on map, 211, 212, 217, 224. 
pole, 22, 47, 152, 153, 289. 
star, 43, 46, 47, 49, 58, 148, 286. 
North Dakota, 89, 90, 234. 
Northfield, Minn., 69, 90. 
North Sea, 187. 
Northwest Territory, survey of, 

228-230. 
Norway, 68, 85, 88. 
Norwood, Richard, 272. 
Nova Scotia, Canada, 82. 
Noumea, New Caledonia, 100. 
Numa, 134. 
Nutation of poles, 288, 318. 



Oblateness of earth, 28-33, 37, 
43, 273-275. 



Obliquity of the ecliptic, 118, 

147, 288, 310, 315. 
Observations of stars, 9. 
Official Railway Guide, 73-75. 
Ogden, Utah, 91. 
Ohio, 90, 230. , 

River, 228. 
Oklahoma, 90. 
Old Farmer's Almanac, 123. 
Old Style, 137-140, 143. 
Olympia, Wash., 91. 
Omaha, Neb., 91. 
Ontario, Canada, 82. 
Oporto, Portugal, 204. 
Opposition, 177, 178, 185. 
Orange River Colony, 82. 
Orbit, of earth, 22, 113, 114, 116- 

119, 122, 132, 147, 152, 246, 

251, 285, 304^306. 
of moon, 177, 178, 262, 288. 
Oregon, 91. 

Origin of geometry, 226. 
Orion (o ri'on). 111. 
Oroya, Peru, 86. 
Orkneys, The, 82. 
Orthographic projection, 190- 

195, 198, 200, 2;i. 
Outlook, The, 76, 142-144. 



Pacific Ocean, 68, 96, 97, 185, 

242. 
Pacific time belt, 68. 
Pago Pago (pron. pango, pango), 

Samoa, 91, 96. 
Palestine, 89. 
Pallas, 293. 

Panama, 69, 85, 87, 89. 
Para, Brazil, 89. 
Parallax, 109, 241, 277, 319. 
Parallelism of earth's axis, 154. 
Parallels, 23, 190-226. 
Paris, France, 28, 29, 41, 55, 64, 

89, 274. 
Parliament, 77. 



INDEX 



333 



Pegasus (pgg'a stts), Square of, 

48. 
Peking, China, 89. 
Pendulum clock, 28, 54, 309. 
Pennsylvania, 78, 90, 228, 229. 
Perigee, 178, 319. 
Perihelion, ll9, 284, 319. 
Peru, 86, 274. 

Pescadores (p6skad6r'ez) Is., 84. 
Phases of the moon, 10, 263. 
Philadelphia, Pa., 26, 27, 59, 67, 

91. 
PhiUppine Is., 87, 101, 102. 
Phobos (fo'bus), 255, 306. 
Phoenicians, 268. 
Photographing, 50. 
Picard (pekar'), Jean, 273, 274. 
Pierre, S. D., 91. 
Pittsburg, Pa., 67, 97. 
Pisces (pis'sez), 294, 296. 
Planetesimal hypothesis, 250. 
Planets, 19, 50, 246, 285, 306. 
Plato, 270. 

Pleiades (ple'yadez), 48, 122. 
Plumb hne, 11, 51, 281, 282. 
Plutarch, 270. 
Point Arena, Cahf., 32. 
Point Barrow, 91. 
Pointing exercise, 38-40. 
Poland, 33, 137. 
Polar diameter of earth, 310. 
Polaris, see polestar. 
Pole, celestial, 46, 170, 171, 284, 
286, 319. 

magnetic, 153. 

nutation of, 288. 

of the echptic, 286, 288. 

terrestrial, 22, 37, 38, 47, 54r- 

56, 60, 61, 152-154, 157, 

193-212, 280, 286, 288, 290. 

Polyconic projection, 222-224. 

Polestar (see North star), 9, 10, 

286. 
Popular Astronomy, 132. 
Portland, Ore., 91. 
Porto Rico, 87, 91. 



Port Said (sa ed'), Egypt, 83. 
Portugal, 41, 86, 89. 
Posidionius (p6s'i do ni fls), 271. 
Practical Navigator, 217. 
Practical work, 300-303. 
Precession of equinoxes, 286- 

288. 
Prince Edward Island, Can., 82. 
Princeton, N. J., 91. 
Principal meridian, 230-236. 
Projectiles, 304r-306. 
Projections, map, 190-225. 
Proofs, form of earth, 24r-29, 

33-35, 274. 
revolution of earth, 104-111, 

277, 278. 
rotation of earth, 51-57, 62, 

107, 155. 
Proper motion of stars, 109. 
Providence, R. I., 91. 
Psalms, 252. 

Ptolemaic system, 276, 319. 
Ptolemy, Claudius, 271. 
Ptolemy Necho, of Egypt, 268. 
Pulkowa, Russia, 73, 86, 89. 
Pythagoras (pi thag'o ras), 269, 

270, 275. 

Q 

Quebec, Canada, 82. 

Queensland, Australia, 82. 

Quito (ke'to), Equador 83, 163, 
174. 

' R 

Radius vector, 284, 319. 

Raleigh, N. C, 91. 

Ranges of townships, 231-233, 
236. 

Rapid City, S. D., 243. 

Rate of curvature of earth's sur- 
face, 27, 28, 43, 44. . 

Refraction of hght, 45, 158, 319. 

Revolution (see Glossary, p. 
320), 104-131, 146, 154, 
184, 246, 248, 251, 254, 257- 
262, 277, 278, 285-289. 



SS4 



INDEX 



Eicher (re shay'), Jean, 28, 29, 

273. 
Richmond, Va., 91. 
Rio de Janeiro, Brazil, 89. 
Rhode Island, 91. 
Rhodesia, Africa, 82. 
Rochester, N. Y., 91. 
Roman calendar, 134, 138. 
Rome, Italy, 73, 89. 
Rotation of earth, 23, 320. 
Rotation, proofs of, 51, 155. 
Rotterdam, Holland, 89. 
Roumania, 68. 
Russia, 32, 86, 89, 96, 102, 103, 

140, 242. 



Sacramento, Calif., 91. 
Sagittarius (sag it ta'ri iis), 296. 
St. John's, Ncv/foundland, 82. 
St. Louis, Mo., 26, 27, 67, 85, 91. 
St. Paul, Minn., 01, 95. 
St. Petersburg, Russia, 73, 86, 

89, 169, 174. 
Salvador (sal va dor'), 86. 
Samoa, 84, 88, 01, 96, 97, 100. 
San Bernardino, Calif., 230. 
San Francisco, Calif., 93, 173. 
San Jose (ho sa'), Costa Rica, 83. 
San Juan del Sur, Nicaragua, 85. 
San Juan, Porto Rico, 91. 
San Rafael (rafagl'), Mexico, 

85. 
San Salvador, Salvador, 86. 
Santiago (san te a'go), Chile, 82. 
Santa Fe, N. M., 91. 
Santo Domingo, 86. 
Saskatchewan, Canada, 82. 
Satellite, see Moon. 
Saturn, 30, 248, 253, 257, 258, 266. 
Savannah, Ga., 91. 
Scale of miles, 195, 216, 224. 
Schott, C. A., 32, 33. 
Scotland, 82, 88, 139, 187. 
Scrap Book, 65. 



Scientific American, 267, 305. 

Scorpio, 296. 

Seasons, 146-175, 168, 169. 

Seattle, Wash., 91. 

Section, 235, 236. 

Seoul (sa ool'), Korea, 85. 

Servia, 68, 86. 

Seven motions of earth, 289. 

Seven ranges of Ohio, 229. 

Sextant, 61. 

Shakespeare, 299. 

Shanghai (shang'hl), China, 83. 

Shetland Is., 82. 

Siam, 88. 

Siberia, 97, 99, 103. 

Sicily, 289. 

Sidereal, clock, 69, 70. 

day, 55, 815. 

month, 177, 318. 

year, 132, 287, 310, 321. 
Signals, time, 71-73, 81-87. 
Signs of zodiac, 116, 294, 320. 
Sines, natural, table of, 311. 
Sirius (sir'ius), 47. 
Sitka, Alaska, 73, 91. 
Snell, Willebrord, 272. 
Solar day, see Day. 
Solar system, 246-267. 

table, 266. 
Solstices, 148, 167, 303, 320. 
Sosigenes (so sig'e nez), 135. 
South America, 34, 168, 186, 

187, 200, 213, 218, 273. 
South AustraUa, 82. 
Southern Cross, 47. 
South Carolina, 90. 
South Dakota, 89-91, 227, 233, 

234,243. 
South, on map, 200, 211, 212, 224. 

pole, 153. 

star, 46-49, 58, 59, 148, 151. 
Spain, 68, 86, 88, 92, 128, 214. 
Spectrograph, 109. 
Spectroscope, 57, 107, 109. 
Sphere, defined, 20. 
Spheroid, 22, 28-33, 320. 



INDEX 



336 



Spitzbergen, 32. 
Spring tides, 185, 188. 
Square of Pegasus, 48. 
Stadium (sta' di um), 271. 
Standard parallel, 233, 234. 
Standard time, 65-88. 
Star, distance of a, 45, 246. 

motions of, 108, 109, 265, 266. 

sun a, 265-267. 
Stereographic projection, 195- 

198, 200, 211. 
Stockholm, Sweden, 86, 89. 
Strabo (stra'bo), 271. 
Strauss, N. M., 76. 
Sun, 10-12, 19, 161, 246-248, 
250, 251, 264-267. 

apparent motions of, 113, 294. 

a star, 265-267. 

declination of, 127, 171-174. 

fast or slow, 62, 123-130. 
Sun Board, 302, 303. 
Sundial, 62, 65, 131, 300-302. 
Survey, 31, 32, 36, 226, 272. 
Surveyor's chain, 226, 310. 
Sweden, 32, 68, 86, 89, 277. 
Switzerland, 68, 86, 137. 
Sydney, Australia, 89. 
Syene, Egypt, 270. 
Symbols, 307. 
Syzygy, 178, 185, 320. 



Tables, list of, 313. 
Tacubaya (ta koo ba'ya), Mex- 
ico, 85. 
Tallahassee, Fla., 91, 230. 
Tamarack mine, 51, 53. 
Tangents, natural, table of, 312. 
Tasmania, 82. 
Taurus, 295, 298. 
Tegucigalpa, Honduras, 84. 
Te egraphjc time signals, 69, 81. 
Tennessee, 90. 
Texas, 89, 90 243. 
Thales (tha'lez), 270. 



Thompson, A. H., 244. 
Thucydides (thu sid'i dez), 132. 
Tidal wave, bore, etc., 185, 189. 
Tides, 176-189, 280, 290-292. 
Tientsin (te gn'tsen), China, 83. 
Tiers of townships, 231, 234, 236. 
Time (see Glossary, p. 320), 
apparent solar, 62. 

ball, 71, 83. 

confusion, 65, 73-76, 144. 

in various countries, 81-87. 

local, 64. 

how determined, 69. 

signals, 69-73, 81-87. 

standard, 65-88, 128, 129. 
Times, London, 139. 
Titicaca, Lake, Peru, 86. 
Todd, David, 106, 289. 
Toga Is., 84. 
Tokyo, Japan, 89. 
Toledo, 0., 73. 
Tonga Is,, 100. 
Toscanellia, 212, 213. 
Township, 228-236. 
Transit instrument, 69. 
Transvaal, 82. 
Trenton, N. J., 91. 
Triangulation, 31, 32, 237,275. 
Tropics, 150, 151, 173, 269, 321. 
Tunis, 83. 

Turkey, 87, 88, 142-144. 
Turkish calendar, 142-144. 
Tutuila (tootwe'la), Samoa, 87, 

96 97. 
Twilight, 161-165. 

U 

Unequal heating, 169, 284. 

United States, 31-34, 36, 42, 
48,, 51, 65, 69, 71, 78, 87, 
89, 97, 98, 101-103, 126, 
128, 129, 160, 168, 187, 224, 
226-236, 289, 294. 

United States Coast and Geo- 
detic Survey, 31, 32, 36, 242, 
244, 245, 310. 



336 



INDEX 



United States Geological Sur- 
vey, 31, 242, 243. 

United States Government Land 
Survey, 226-236. 

United States Naval Observa- 
tory, 69-73, 81. 

University of California, 155. 

University of Chicago, 250, 302. 

Ur, ancient Chaldean city, 141. 

Uruguay, 87, 89. 

Uranus (ti'ra nus), 253, 258, 
259, 266. 

Ursa Major, 9. 

Utah, 91, 243. 

V 

Valparaiso (val pa ri'so), Chile, 

82, 89. 
Van der Grinten, Alphons, 208. 
Vibrations, color, 106, 107. 
Victoria, Australia, 82, 89. 
Virginia City, Nev., 91. 
Virgo, 295. 

Velocity of rotation, 58. 
Venezuela, 87. 

Venus, 183, 253, 255, 266, 277. 
Vernal equinox, see Equinox. 
Vertical ray of sun, 146, 147, 

152, 155,156, 165,166,313. 
Vineocaya (vin ko ka'ya), Peru, 

164. 
Virginia, 91. 
Voltaire, 274. 
Volume of earth, 310. 

W 

Wady-Haifa (wa'de hal'fa), 

Egypt, 83. 
Wallace, Kan., 35. 
Wandering of the poles, 288. 



Washington, 91. 

Washington, D. C, 35, 42, 67, 
71, 72, 85, 86, 91, 124, 230. 

Washington, George, 138, 139. 

Watch, to set by sun, 129. 

Weight, see Gravity. 

Wellington, New Zealand, 73. 

Western European time, 68. 

West Virginia, 91. 

What keeps the members of the 
solar system in their or- 
bits? 304r-306. 

Wheeling, W. Va., 91. 

Wilhelm II., Emperor, 77. 

Wilmington, Del., 91. 

Winona, Minn., 91. 

Winter constellations, 111. 

Wisconsin, 67, 89, 90, 103, 231. 

Woodward, R. S., 5, 35. 

World Almanac, 123. 

Wyoming, 90. 

X 

Xico, Mexico, 85. 

Y 

Yaeyama (ye ya'ma) Is., 84. 
Yaqui River, Mexico, 85. 
Year, 132, 133, 287, 310, 321. 
Young, C. A., 163. 
Youth's Companion, 72, 153, 

165. 
Ysleta (is la'ta), Tex., 76. 



Zikawei (zi ka'we), China, 83. 
Zodiac, 116, 117, 141, 293, 322. 
Zones, 152, 254-258, 260, 263. 



\